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+Incentive Compatibility in the Auto-bidding World
+Yeganeh Alimohammadi†, Aranyak Mehta∗ and Andres Perlroth∗
+February 1, 2023
+Abstract
+Auto-bidding has recently become a popular feature in ad auctions. This feature enables
+advertisers to simply provide high-level constraints and goals to an automated agent, which
+optimizes their auction bids on their behalf. These auto-bidding intermediaries interact in
+a decentralized manner in the underlying auctions, leading to new interesting practical and
+theoretical questions on auction design, for example, in understanding the bidding equilibrium
+properties between auto-bidder intermediaries for different auctions. In this paper, we examine
+the effect of different auctions on the incentives of advertisers to report their constraints to the
+auto-bidder intermediaries. More precisely, we study whether canonical auctions such as first
+price auction (FPA) and second price auction (SPA) are auto-bidding incentive compatible (AIC):
+whether an advertiser can gain by misreporting their constraints to the autobidder.
+We consider value-maximizing advertisers in two important settings: when they have a budget
+constraint and when they have a target cost-per-acquisition constraint. The main result of our
+work is that for both settings, FPA and SPA are not AIC. This contrasts with FPA being AIC
+when auto-bidders are constrained to bid using a (sub-optimal) uniform bidding policy. We
+further extend our main result and show that any (possibly randomized) auction that is truthful
+(in the classic profit-maximizing sense), scalar invariant and symmetric is not AIC. Finally, to
+complement our findings, we provide sufficient market conditions for FPA and SPA to become
+AIC for two advertisers. These conditions require advertisers’ valuations to be well-aligned. This
+suggests that when the competition is intense for all queries, advertisers have less incentive to
+misreport their constraints.
+From a methodological standpoint, we develop a novel continuous model of queries. This
+model provides tractability to study equilibrium with auto-bidders, which contrasts with the
+standard discrete query model, which is known to be hard. Through the analysis of this model,
+we uncover a surprising result: in auto-bidding with two advertisers, FPA and SPA are auction
+equivalent.
+†Stanford University, yeganeh@stanford.edu
+∗Google, {aranyak,perlroth}@google.com
+1
+arXiv:2301.13414v1  [econ.TH]  31 Jan 2023
+
+1
+Introduction
+Auto-bidding has become a popular tool in modern online ad auctions, allowing advertisers to
+set up automated bidding strategies to optimize their goals subject to a set of constraints. By
+using algorithms to adjust the bid for each query, auto-bidding offers a more efficient and effective
+alternative to the traditional fine-grained bidding approach, which requires manual monitoring and
+adjustment of the bids.
+There are three main components in auto-bidding paradigm: 1) the advertisers who provide
+high-level constraints to the auto-bidders, 2) the auto-bidder agents who bid – in a decentralized
+manner – on behalf of each advertiser to maximize the advertiser’s value subject to their constraints,
+and 3) the query-level auctions where queries are sold (see Figure 1).
+Figure 1: The Auto-bidding Process: Advertisers submit constraints and receive query allocations
+with specified costs as output. Inside the auto-bidding feature, each advertiser has an agent that
+optimizes bidding profile within each advertiser’s constraints.
+Current research has made important progress in studying the interactions of the second and third
+components in the auto-bidding paradigm, particularly in understanding equilibrium properties (e.g.,
+welfare and revenue) between the auto-bidders intermediaries for different auction rules (Aggarwal
+et al., 2019; Balseiro et al., 2021a; Deng et al., 2021a; Mehta, 2022; Liaw et al., 2022). There is also
+work on mechanism design for this setting in more generality, i.e., between the advertisers and the
+auctioneer directly abstracting out the second component (Balseiro et al., 2021c, 2022; Golrezaei
+et al., 2021b).
+Our work, instead, examines the relation between value-maximizing advertisers, who maximize
+the value they obtain subject to a payment constraint, and the other two components of the auto-
+bidding paradigm. More precisely, we study the impact of different auction rules on the incentives
+of advertisers to report their constraints to the auto-bidder intermediaries. We specifically ask
+whether canonical auctions such as first price auction (FPA), second price auction (SPA) and general
+truthful auctions are auto-bidding incentive compatible (AIC) - in other words, can advertisers gain
+by misreporting their constraints to the auto-bidder?
+We consider value-maximizing advertisers in two important settings: when they have a budget
+constraint and when they have a target cost-per-acquisition (tCPA) constraint1. The main result of
+1The former is an upper bound on the total spend, and the latter is an upper bound on the average spend per
+acquisition (sale). Our results clearly hold for more general autobidding features, such as target return on ad-spend
+(tRoAS) where the constraint is an upper bound on the average spend per value generated.
+1
+
+Auto-
+constraints
+bids
+Advertiser
+bidder
+Agent
+Auto-
+constraints
+bids
+Auction
+Advertiser
+bidder
+per query
+Agent
+Auto-
+constraints
+bids
+Advertiser
+bidder
+Agentour work is that for both settings, FPA and SPA are not AIC. This contrasts with FPA being AIC
+when auto-bidders are constrained to bid using a (sub-optimal) uniform bidding policy. We further
+generalize this surprising result and show that any (possibly randomized) truthful auction having a
+scale invariance and symmetry property is also not AIC. We complement our result by providing
+sufficient market conditions for FPA and SPA to become AIC for two advertisers. These conditions
+require advertisers’ valuations to be well-aligned. This suggests that when the competition is intense
+for all queries, advertisers have less incentive to misreport their constraints.
+In our model, each advertiser strategically reports a constraint (either a tCPA or a budget) to
+an auto-bidder agent which bids optimally on their behalf in each of the queries. Key in our model,
+we consider a two stage game where first advertisers submit constraints to the auto-bidders and, in
+the subgame, auto-bidders reach a bidding equilibrium across all query-auctions. Thus, when an
+advertiser deviates and reports a different constraint to its auto-bidder, the whole bidding subgame
+equilibrium can change.2 In this context, an auction rule is called auto-bidding incentive compatible
+(AIC) if, for all equilibria, it is optimal for the advertiser to report their constraint to the auto-bidder.
+1.1
+Main Results
+We begin our results by presenting a stylized example in Section 2 that demonstrates how auto-
+bidding with SPA is not AIC (Theorem 2.1). Our example consists of a simple instance with three
+queries and two advertisers. This example highlights a scenario where an advertiser can benefit from
+lowering their reported budget or tCPA-constraint.
+We then introduce a continuous query model that departs from the standard auto-bidding model
+by considering each query to be of infinitesimal size. This model provides tractability in solving
+equilibrium for general auction rules like FPA which is key to study the auto-bidding incentive
+compatibility properties of such auctions. Further, this continuous-query model succinctly captures
+real-world scenarios where the value of a single query is negligible compared to the pool of all queries
+that are sold.
+Under the continuous-query model, we study the case where queries are sold using FPA and show
+that in the auto-bidding paradigm, FPA is not AIC (Section 4). We first characterize the optimal
+bidding strategy for each auto-bidder agent which, surprisingly, has a tractable form.3 We then
+leverage this tractable form to pin down an equilibrium for the case of two auto-bidders when both
+auto-bidders either face a budget or tCPA constraint. In this equilibrium, queries are divided between
+the two advertisers based on the ratio of their values for each advertiser. Specifically, advertiser 1
+receives queries for which the ratio of its value to the other advertiser’s value is higher than a certain
+threshold. From this point, determining the equilibrium reduces to finding a threshold that make
+advertisers’ constraints tight (see Lemma 4.4 for more detail). We then show that for instances where
+the threshold lacks monotonicity with the auto-bidders constraints, advertisers have an incentive
+to misreport the constraint to the auto-bidder (Theorem 4.1). Conversely, when the thresholds
+are monotone advertisers report constraints truthfully. We show conditions on the advertisers’
+valuations, for the two-advertisers setting, to guarantee this monotonicity (Theorem 4.10). This
+condition requires a strong positive correlation of the advertisers’ valuations across the queries. As a
+2This two stage model captures the idea that auto-bidding systems rapidly react to any change in the auction.
+Hence, if there is any change in the bidding landscape, auto-bidders quickly converge to a new equilibrium.
+3Notice that in the discrete-query model, there is no simple characterization for the auto-bidder best response in a
+FPA.
+2
+
+practical insight, our results suggest that for settings where the competition on all queries is intense,
+advertisers’ incentives to misreport is weak.
+We then explore the case where, in FPA, auto-bidders are constrained to bid using a uniform
+bidding strategy: the bid on each query is a constant times the advertiser’s value for the query.4
+Uniform bidding is only an optimal strategy when auctions are truthful (Aggarwal et al., 2019).
+Even though for FPA these strategies are suboptimal, they have gained recent attention in the
+literature due to their tractability Conitzer et al. (2022a,b); Chen et al. (2021); Gaitonde et al. (2022).
+We show that in such a scenario, FPA with uniform bidding turns out to be AIC (Theorem 4.2).
+However, we note that while this proves AIC in our model, the suboptimality of uniform bidding for
+FPA can give rise to incentives to deviate in other ways outside our model, e.g., by splitting the
+advertising campaigns into multiple campaigns with different constraints. These considerations are
+important when implementing this rule in practice.
+The second part of the paper pivots to the case where auctions are truthful, that is, auctions in
+which it is optimal for a profit-maximizing agent to bid their value. We first study the canonical
+SPA and show that, in our continuous-query model, SPA and FPA are auction equivalent. That is,
+the allocation and payments among the set of reasonable equilibria (Theorem 5.5).5 As a Corollary,
+the results we obtain for FPA apply to SPA as well: SPA is not AIC and; and we derive sufficient
+conditions on advertisers’ valuations so that SPA is AIC for two advertisers. We then consider a
+general class of randomized truthful auctions. We show that if the allocation rule satisfies these
+natural conditions:6 (i) scaled invariant (if all bids are multiplied by the same factor then the
+allocation doesn’t change), and (ii) is symmetric (bidders are treated equally); then the auction rule
+is not AIC.
+The main results of the paper are summarized in Table 1.
+Per Query Auction
+AIC
+Second-Price Auction
+Not AIC
+Truthful Auctions
+Not AIC
+First-Price Auction
+Not AIC
+First-Price Auction with Uniform Bidding
+AIC7
+Table 1: Main Results
+1.2
+Related Work
+The study of auto-bidding in ad auctions has gained significant attention in recent years. One of the
+first papers to study this topic is Aggarwal et al. (2019), which presents a mathematical formulation
+for the auto-bidders problem given a fixed constraints reported by advertisers. They show that
+uniform bidding is an optimal strategy if and only if auctions are truthful (in the profit-maximizing
+sense). They further started an important line of work to measure, using a Price of Anarchy (PoA)
+approach, the welfare implications when auto-bidders are bidding in equilibrium for different auctions.
+4Uniform bidding strategy is also known in the literature as pacing bidding Conitzer et al. (2022a); Chen et al.
+(2021); Conitzer et al. (2022b); Gaitonde et al. (2022).
+5We show the auction equivalence among uniform bidding equilibria for SPA and threshold-type equilibrium for
+FPA.
+6These conditions have been widely studied in the literature due to their practical use (Mehta, 2022; Liaw et al.,
+2022; Allouah and Besbes, 2020).
+7As previously discussed, implementing FPA with the suboptimal uniform bidding policy can create other distortion
+on advertisers’ incentives (e.g., splitting their campaign into multiple campaigns with different constraints).
+3
+
+Current results state that for SPA the PoA is 2 Aggarwal et al. (2019) and also for FPA Liaw et al.
+(2022)8, and, interestingly, it can be improved if the auction uses a randomized allocation rule Mehta
+(2022); Liaw et al. (2022). In a similar venue, Deng et al. (2021b); Balseiro et al. (2021b) studies
+models where the auction has access to extra information and show how reserves and boosts can be
+used to improve welfare and efficiency guarantees.
+A second line of work, studies how to design revenue-maximizing auctions when bidders are
+value-maximizing agents and may have private information about their value or their constraints
+(Golrezaei et al., 2021b; Balseiro et al., 2021c,b). In all these settings, the mechanism designer is not
+constrained to the presence of the auto-bidding intermediaries (Component 2 in Figure 1). Our study
+has added structure by having advertisers submit their constraints first, followed by a decentralized
+subgame to achieve a bidding equilibrium before allocating and determining payments. Thus, a priori
+their mechanism setting can achieve broader outcomes than in our auto-bidding constraint paradigm.
+Interestingly, for the one query case the authors show that FPA with a uniform bidding policy is
+optimal Balseiro et al. (2021c). Our results complement theirs and show that such mechanism is
+implementable in auto-bidding constraint paradigm and is AIC.
+Closer to our auto-bidding paradigm, a recent line of work has started to study the incentive of
+advertisers when bidding via an auto-bidder intermediary. Mehta and Perlroth (2023) show that a
+profit-maximizing agent may benefit by reporting a target-based bidding strategy to the auto-bidder
+when the agent has concern that the auctioneer may change (ex-post) the auction rules. Also, in
+an empirical work, Li and Tang (2022) develop a new methodology to numerically approximate
+auto-bidding equilibrium and show numerical examples where advertisers may benefit my reporting
+their constraints on SPA. Our work complements their findings by showing under a theoretical
+framework that SPA is not AIC.
+Our work also connects with the literature about auction with budgeted constraint bidders. In
+particular, our results are closely related to Conitzer et al. (2022a) who study FPA with uniform
+bidding (a.k.a. pacing bidding). They introduce the concept of the first-price auction pacing
+equilibrium (FPPE) for budget-constrained advertisers, which is the same as the equilibrium in our
+auto-bidding subgame. They show that in FPPE the revenue and welfare are monotone increasing
+as a function of the advertisers’ budgets. In our work, we show that in FPPE, advertisers’ values
+are monotone as a function of their reported budget. In addition, they differentiate between first
+and second-price by showing that FPPE is computable, unlike SPPE, where maximizing revenue
+has previously been known to be NP-hard Conitzer et al. (2022b), and that the general problem
+of approximating the SPPE is PPAD-complete Chen et al. (2021). In contrast, we show in the
+continuous model both SPA and FPA are tractable. Interestingly, this dichotomy between FPA and
+SPA (both with uniform bidding) is reflected in our work as well – the former is AIC, while the
+latter is not.
+Uniform bidding has been explored in a separate body of research on repeated auctions, without
+the presence of auto-bidding. Balseiro and Gur (2019) investigate strategies to minimize regret in
+simultaneous first-price auctions with learning. Gaitonde et al. (2022) take this concept further by
+extending the approach to a wider range of auction settings. Furthermore, Golrezaei et al. (2021a)
+examines how to effectively price and bid for advertising campaigns when advertisers have both ROI
+and budget constraints.
+8The authors show that for a general class of deterministic auctions PoA ≥ 2.
+4
+
+2
+Warm Up: Second Price Auction is not AIC!
+To understand the implications of the auto-bidding model, we start with an example of auto-bidding
+with the second-price auction. Through this example, we will demonstrate the process of determining
+the equilibrium in an auto-bidding scenario and emphasize a case where the advertiser prefers to
+misreport their budget leading to the following theorem.
+Theorem 2.1. For the budget setting (when all advertisers are budgeted-constrained) and for the
+tCPA-setting (when all advertisers are tCPA-constrained), we have that SPA is not AIC. That is,
+there are some instances where an advertiser benefits by misreporting its constraint.
+Proof. Consider two budget-constrained advertisers and three queries Q = {q1, q2, q3}, where the
+expected value of winning query q for advertiser a is denoted by va(q), and it is publicly known (as
+in Table 2). At first, each advertiser reports their budget to the auto-bidder B1 = 2, and B2 = 4.
+Then the auto-bidder agents, one for each advertiser, submit the bidding profiles (to maximize their
+advertisers’ value subject to the budget constraint). The next step is a second-price auction per
+query, where the queries are allocated to the highest bidder.
+q1
+q2
+q3
+Advertiser 1
+4
+3
+2
+Advertiser 2
+1
+1.3
+10
+Table 2: SPA with two budget constraint advertisers is not AIC: The value of each query for each
+advertiser.
+Finding the equilibrium bidding strategies for the auto-bidder agents is challenging, as the
+auto-bidder agents have to find the best-response bids with respect to the other auto-bidder agents,
+and each auto-bidder agent’s bidding profile changes the cost of queries for the rest of the agents.
+To calculate such an equilibrium between auto-bidder agents, we use the result of Aggarwal et al.
+(2019) to find best-response strategies. Their result states that the best response strategy in any
+truthful auto-bidding auction is uniform bidding.9 In other words, each agent optimizes over one
+variable, a bidding multiplier µa, and then bids on query q with respect to the scaled value µava(q).
+We show that with the given budgets B1 = 2 and B2 = 4, an equilibrium exists such that
+advertiser 1 only wins q1, and µ1 = 0.5 and µ2 = 1 result in such an equilibrium. To this end,
+we need to check: 1) Allocation: with bidding strategies b1 = (µ1v1(q1), µ1v1(q2), µ1v1(q3)) and
+b2 = (µ2v2(q1), µ2v2(q2), µ2v2(q3)), advertiser 1 wins q1 and advertiser 2 wins q2 and q3, 2) Budget
+constraints are satisfied, and 3) Bidding profiles are the best response: The auto-bidder agents do
+not have the incentive to increase their multiplier to get more queries. These three conditions are
+checked as follows:
+1. Allocation inequalities: For each query, the advertiser with the highest bid wins it.
+v1(q1)
+v2(q1) ≥ µ2
+µ1
+= 1
+0.5 ≥ v1(q2)
+v2(q2) ≥ v1(q3)
+v2(q3).
+2.
+Budget constraints: Since the auction is second-price the cost of query q for advertiser 1 is
+µ2v2(q) and for advertiser 2 is µ1v1(q). So, we must have the following inequalities to hold so
+9They show uniform bidding is almost optimal, but in Appendix A we show that in this example it is exactly
+optimal.
+5
+
+that the budget constraints are satisfied:
+2 = B1 ≥ µ2v2(q1) = 1
+(Advertiser 1),
+4 = B2 ≥ µ1(v1(3) + v1(q2)) = 2.5
+(Advertiser 2).
+3. Best response: Does the advertiser’s agent have incentive to raise their multiplier to get more
+queries? If not, they shouldn’t afford the next cheapest query.
+2 < µ2(v2(q1) + v2(q2)) = 2.3
+(Advertiser 1),
+4 < µ1(v1(q3) + v1(q2) + v1(q1)) = 4.5
+(Advertiser 2).
+Since all the three conditions are satisfied. Thus, this profile is an equilibrium for th auto-bidders
+bidding game. In this equilibrium, advertiser 1 wins q1 and advertiser 2 wins q2 and q3.
+Now, consider the scenario that advertiser 1 wants to strategically report their budget B1 to the
+auto-bidder. Suppose the first advertiser decreases their budget. Intuitively, the budget constraint
+for the auto-bidder agent should be harder to satisfy, and hence the advertiser should not win more
+queries. But, contrary to this intuition, when advertiser 1 reports a lower budget B′
+1 = 1, we show
+that, given the unique auto-bidding equilibrium, advertiser 1 wins q1 and q2 (more queries than the
+case where advertiser 1 reports B1 = 2). Similar to above, we can check that µ′
+1 =
+1
+2.3, and µ′
+2 = 1
+results in an equilibrium (we prove the uniqueness in Appendix A):
+1. Allocation: advertiser 1 wins q1 and q2 since it has a higher bid on them,
+v1(q1)
+v2(q1) ≥ v1(q2)
+v2(q2) ≥ µ′
+2
+µ′
+1
+= 1
+2.3 ≥ v1(q3)
+v2(q3).
+2. Budget constraints:
+4 ≥ v1(q3),
+and
+1 = (1/2.3)(v2(q1) + v2(q2)).
+3. Best response:
+4 < 1(v1(q3) + v1(q2)),
+and
+1 < (1/2.3)(v2(q1) + v2(q2) + v2(q3)).
+This surprising example leads to the first main result of the paper. In Appendix A, we will
+generalize the above example to the case of tCPA-constrained advertisers with the same set of queries
+as in Table 2.
+Before studying other canonical auctions, in the next section we develop a tractable model of
+continuous query. Under this model it turns out that the characterization of the auto-bidders bidding
+equilibria when the auction is not SPA is tractable. This tractability is key for studying auto-bidding
+incentive compatibility.
+6
+
+3
+Model
+The baseline model consists of a set of A advertisers competing for q ∈ Q single-slot queries owned by
+an auctioneer. We consider a continuous-query model where Q = [0, 1]. Let xa(q) be the probability
+of winning query q for advertiser a. Then the expected value and payment of winning query q at price
+pa(q) are xa(q)va(q)dq and pa(q)dq.10, 11 Intuitively, this continuous-query model is a first-order
+approximation for instances where the size of each query relative to the whole set is small.
+The auctioneer sells each query q using a query-level auction which induces the allocation and
+payments (xa(q), pa(q))a∈A as a function of the bids (ba)a∈A. In this paper, we focus on the First
+Price Auction (FPA), Second Price Auction (SPA) and more generally any Truthful Auction (see
+Section 5.2 for details).
+Auto-bidder agent:
+Advertisers do not participate directly in the auctions, rather they report high-level goal constraints
+to an auto-bidder agent who bids on their behalf in each of the queries. Thus, Advertiser a reports a
+budget constraint Ba or a target cost-per-acquisition constraint (tCPA) Ta to the auto-bidder. Then,
+the auto-bidder taking as fixed other advertiser’s bid, submits bids ba(q) to induce xa(q), pa(q) that
+solves
+max
+� 1
+0
+xa(q)va(q)dq
+(1)
+s.t.
+� 1
+0
+pa(q)dq ≤ Ba + Ta
+� 1
+0
+xa(q)va(q)dq.
+(2)
+The optimal bidding policy does not have a simple characterization for a general auction. However,
+when the auction is truthful (like SPA) the optimal bid take a simple form in the continuous model.
+(Aggarwal et al., 2019).
+Remark 3.1 (Uniform Bidding). If the per-query auction is truthful, then uniform bidding is the
+optimal policy for the autobidder. Thus, ba(q) = µ · va(q) for some µ > 0. We formally prove this in
+Claim 5.4.
+Advertisers
+Following the current paradigm in autobidding, we consider that advertisers are value-maximizers
+and can be two of types: a budget-advertiser or tCPA-advertiser. Payoffs for these advertisers are as
+follows.
+• For a budget-advertiser with budget Ba, the payoff is
+ua =
+�� 1
+0 xa(q)va(q)dq
+if
+� 1
+0 pa(q)dq ≤ Ba
+−∞
+if not.
+10All functions va, xa, pa are integrable with respect to the Lebesgue measure dq.
+11The set Q = [0, 1] is chosen to simplify the exposition. Our results apply to a general metric measurable space
+(Q, A, λ) with atomless measure λ.
+7
+
+• For a tCPA-advertiser with target Ta, the payoff is
+ua =
+�� 1
+0 xa(q)va(q)dq
+if
+� 1
+0 pa(q)dq ≤ Ta ·
+� 1
+0 xa(q)va(q)dq
+−∞
+if not.
+Game, Equilibrium and Auto-bidding Incentive Compatibility (AIC)
+The timing of the game is as follows. First, each advertiser depending on their type submits a
+budget or target constraint to an auto-bidder agent. Then, each auto-bidder solves Problem 1 for
+the respective advertiser. Finally, the per-query auctions run and allocations and payments accrue.
+We consider a complete information setting and use subgame perfect equilibrium (SPE) as
+solution concept. Let Va(B′
+a; Ba) the expected payoff in the subgame for a budget-advertiser with
+budget Ba that reports B′
+a to the auto-bidder (likewise we define Va(T ′
+a; Ta) for the tCPA-advertiser).
+Definition 3.2 (Auto-bidding Incentive Compatibility (AIC)). An auction rule is Auto-bidding
+Incentive Compatible (AIC) if for every SPE we have that Va(Ba; Ba) ≥ Va(B′
+a; Ba) and Va(Ta; Ta) ≥
+Va(T ′
+a; Ta) for every Ba, B′
+a, Ta, T ′
+a.
+Similar to classic notion of incentive compatibility, an auction rule satisfying AIC makes the
+advertisers’ decision simpler: they simply need to report their target to the auto-bidder. However,
+notice that the auto-bidder plays a subgame after advertiser’s reports. Thus, when Advertiser a
+deviates and submit a different constraint, the subgame outcome may starkly change not only on
+the bids of Advertiser a but also other advertisers bid may change as well.
+4
+First Price Auctions
+In this section, we demonstrate that the first price auction is not auto-bidding incentive compatible.
+Theorem 4.1. Suppose that there are at least two budget-advertisers or two tCPA-advertisers, then
+FPA is not AIC.
+Later in Section 4.2, we show a complementary result by providing sufficient conditions on
+advertisers’ value functions to make FPA be AIC for the case of two advertisers. We show that this
+sufficient condition holds in many natural settings, suggesting that in practice FPA tends to be AIC.
+Then in Section 4.3, we turn our attention to FPA where autobidders are restricted to use
+uniform bidding across the queries. In this case, we extend to our continuous-query model the result
+of Conitzer et al. (2022a) and show the following result.
+Theorem 4.2. FPA restricted to uniform bidding is AIC.
+4.1
+Proof of Theorem 4.1
+We divide the proof of Theorem 4.1 in three main steps. Step 1 characterizes the best response
+bidding profile for an autobidder in the subgame. As part of our analysis, we derive a close connection
+between first and second price auctions in the continuous-query model that simplifies the task of
+finding the optimal bidding for each query to finding a single multiplying variable for each advertiser.
+In Step 2, we leverage the tractability of our continuous-query model and pin-down the subgame
+bidding equilibrium when there are either two budget-advertisers or two tCPA-advertisers in the
+8
+
+game (Lemma 4.4). We derive an equation that characterizes the ratio of the multipliers of each
+advertiser as a function of the constraints submitted by the advertisers. This ratio defines the set of
+queries that each advertiser wins, and as we will see the value accrued by each advertiser is monotone
+in this ratio. So, to find a non-AIC example, one has to find scenarios where the equilibrium ratio is
+not a monotone function of the input constraints which leads to the next step.
+To conclude, we show in Step 3 an instance where the implicit solution for the ratio is nonmono-
+tonic, demonstrating that auto-bidding in first-price auctions is not AIC. As part of our proof, we
+interestingly show that AIC is harder to satisfy when advertisers face budget constraints rather than
+tCPA constraints (see Corollary 4.6).
+Step 1: Optimal Best Response
+The following claim shows that, contrary to the discrete-query model, the best response for the
+autobidder in a first price auction can be characterized as function of a single multiplier.
+Claim 4.3. Taking other auto-bidders as fixed, there exists a multiplier µa ≥ 0 such that the following
+bidding strategy is optimal:
+ba(q) =
+�
+maxa′̸=a(ba′(q))
+µava(q) ≥ maxa′̸=a(ba′(q))
+0
+µava(q) ̸= maxa′(ba′(q)).
+The result holds whether the advertiser is budget-constrained or tCPA-constrained12.
+Proof. We show that in a first-price auction, the optimal bidding strategy is to bid on queries with
+a value-to-price ratio above a certain threshold. To prove this, we assume that the bidding profile of
+all advertisers is given. Since the auction is first-price, advertiser a can win each query q by fixing
+small enough ϵ > 0 and paying maxa′̸=a(ba′(q)) + ϵ. So, let pa(q) = maxa′̸=a(ba′(q)), be the price of
+query q. Since we have assumed that the value functions of all advertisers are integrable (i.e., there
+are no measure zero sets of queries with a high value), in the optimal strategy pa is also integrable
+since it is suboptimal for any advertiser to bid positive (and hence have a positive cost) on a measure
+zero set of queries.
+First, consider a budget-constrained advertiser. The main idea is that since the prices are
+integrable, the advertiser’s problem is similar to a continuous knapsack problem. In a continuous
+knapsack problem, it is well known that the optimal strategy is to choose queries with the highest
+value-to-cost ratio Goodrich and Tamassia (2001). Therefore, there must exist a threshold, denoted
+as µ, such that the optimal strategy is to bid on queries with a value-to-price ratio of at least µ. So
+if we let µa = 1
+µ, then advertiser a bids on any query with µava(q) ≥ pa(q).
+We prove it formally by contradiction. Assume to the contrary, that there exist non-zero measure
+sets X, Y ⊂ Q such that for all x ∈ X and y ∈ Y , the fractional value of x is less than the fractional
+value of y, i.e.,
+va(x)
+pa(qx) < va(y)
+pa(y), and in the optimal solution advertiser a gets all the queries in X and
+no query in Y . However, we show that by swapping queries in X with queries in Y with the same
+price, the advertiser can still satisfy its budget constraint while increasing its value.
+To prove this, fix 0 < α < min(
+�
+X pa(q)dq,
+�
+Y pa(q)dq). Since the Lebesgue measure is atomless,
+there exists subsets X′ ⊆ X and Y ′ ⊆ Y such that α =
+�
+X′ pa(q)dq =
+�
+Y ′ pa(q)dq. Since the value
+12In FPA ties are broken in a way that is consistent with the equilibrium. This is similar to the pacing equilibrium
+notion where the tie-breaking rule is endogenous to the equilibrium Conitzer et al. (2022a).
+9
+
+per cost of queries in Y is higher than queries in X, by swapping queries of X′ with Y ′, the value
+of the new sets increases, while the cost does not change. Therefore, the initial solution cannot be
+optimal.
+A similar argument holds for tCPA-constrained advertisers. Swapping queries in X′ with Y ′ does
+not change the cost and increases the upper bound of the tCPA constraint, resulting in a feasible
+solution with a higher value. Therefore, the optimal bidding strategy for tCPA constraint is also
+ba(q) as defined in the statement of the claim.
+Step 2: Equilibrium Characterization
+The previous step showed that the optimal bidding strategy is to bid on queries with a value-to-price
+ratio above a certain threshold. Thus, we need to track one variable per auto-bidder to find the
+subgame equilibrium.
+In what follows, we focus on the case of finding the variables when there are only two advertisers in
+the game. This characterization of equilibrium gives an implicit equation for deriving the equilibrium
+bidding strategy, which makes the problem tractable in our continuous-query model.13.
+From Claim 4.3 we observe that the ratio of bidding multipliers is key to determine the set of
+queries that each advertiser wins. To map the space of queries to the bidding space, we introduce
+the function h(q) = v1(q)
+v2(q). Hence, for high values of h, the probability that advertiser 1 wins the
+query increases. Also, notice that without loss of generality, we can reorder the queries on [0, 1] so
+that h is non-decreasing.
+In what follows, we further assume that h is increasing on [0, 1]. This implies that h is invertible
+and also differentiable almost everywhere on [0, 1]. With these assumptions in place, we can now
+state the following lemma to connect the subgame equilibrium to the ratio of advertisers’ values.
+Lemma 4.4. [Subgame Equilibrium in FPA] Given two budget-constrained auto-bidders with budget
+B1 and B2, let µ1 and µ2 be as defined in Claim 4.3 for auto-bidding with FPA. Also assume that
+h(q) = v1(q)
+v2(q) as defined above is strictly monotone. Then µ1 =
+B2
+E[z1(z≥r)] and µ2 = µ1r, where r is
+the solution of the following implicit function,
+rE[1[z ≥ r)]
+E[z1(z ≤ r)] = B1
+B2
+.
+(3)
+Here, E[·] is defined as E[P(z)] =
+� ∞
+0 P(z)f(z)dz, where f(z) = v2(h−1(z))
+h′(h−1(z)) wherever h′ is defined,
+and it is zero otherwise.
+Also, for two tCPA auto-bidders with targets T1 and T2, we have µ1 = T1E[1(z≤r)]
+E[1(z≥r)]
+and µ2 = µ1r,
+where r is the answer of the following implicit function,
+rE[1(z ≥ r)]
+E[z1(z ≥ r)]
+E[1(z ≤ r)]
+E[z1(z ≤ r)] = T1
+T2
+.
+(4)
+Remark 4.5. The function f intuitively represents the expected value of the queries that advertiser
+2 can win as well as the density of the queries that advertiser 1 can win. Also, the variable r shows
+the cut-off on how the queries are divided between the two advertisers. In the proof, we will see that
+the advertisers’ value at equilibrium is computed with respect to f: Advertiser 1’s overall value is
+� ∞
+r
+zf(z)dz and advertiser 2’s overall value is
+� r
+0 f(z)dz.
+13Notice that for the discrete-query model finding equilibrium is PPAD hard Filos-Ratsikas et al. (2021)
+10
+
+Proof. First, consider budget constraint auto-bidders. Given Claim 4.3, in equilibrium price of query
+q is min(µ1v1(q), µ2v2(q)). Therefore, the budget constraints become:
+B1 =
+� 1
+0
+µ2v2(q)1(µ2v2(q) ≤ µ1v1(q))dq,
+B2 =
+� 1
+0
+µ1v1(q)1(µ2v2(q) ≥ µ1v1(q))dq.
+With a change of variable from q to z = h(q) and letting r = µ2
+µ1 , we have:
+B1 =
+� ∞
+r
+µ2v2(h−1(z))dh−1(z)
+dz
+dz
+B2 =
+� r
+0
+µ1v1(h−1(z))dh−1(z)
+dz
+dz.
+Observe that v1(h−1(z)) = zv2(h−1(z)), then if we let f(z) = v2(h−1)(h−1)′ =
+v2(h−1(z))
+h′(h−1(z)), the
+constraints become
+B1 = µ2
+� ∞
+r
+f(z)dz,
+(5)
+B2 = µ1
+� r
+0
+zf(z)dz.
+(6)
+We obtain Equation (3) by diving both sides of Equation (5) by the respective both sides of
+Equation (6).
+Now, consider two tCPA constrained auto-bidders. Similar to the budget-constrained auto-bidders,
+we can write
+T1
+� 1
+0
+v1(q)1(µ2v2(q) ≤ µ1v1(q))dq =
+� 1
+0
+µ2v2(q)1(µ2v2(q) ≤ µ1v1(q))dq
+T2
+� 1
+0
+v2(q)1(µ2v2(q) ≥ µ1v1(q))dq =
+� 1
+0
+µ1v1(q)1(µ2v2(q) ≥ µ1v1(q))dq
+The same way of changing variables leads to the following:
+T1
+T2
+� ∞
+r
+xf(x)dx
+� r
+0 f(x)
+= r
+� ∞
+r
+f(x)dx
+� r
+0 xf(x)dx .
+This finishes the proof of the lemma.
+The previous theorem immediately implies that any example of valuation functions that is
+non-AIC for budget-advertisers, it will be non-AIC for tCPA-advertisers as well.
+Corollary 4.6. If auto-bidding with the first-price and two budget-advertisers is not AIC, then
+auto-bidding with the same set of queries and two tCPA-advertisers is also not AIC.
+11
+
+Proof. Recall that advertiser 1 wins all queries with h(q) ≥ r. So, the value accrued by advertiser
+1 is decreasing in r. So, if an instance of auto-bidding with tCPA-constrained advertisers is not
+AIC for advertiser 1, then the corresponding function r
+� ∞
+r
+f(x)dx
+� r
+0 xf(x)dx
+� r
+0 f(x)
+� ∞
+r
+xf(x)dx (same as (4)) must be
+increasing for some r′.
+On the other hand, recall that r
+� ∞
+r
+f(x)dx
+� r
+0 xf(x)dx is the ratio for budget-constrained bidders equilibrium
+as in (3). The additional multiplier in the equilbirum equation of tCPA constraint advertiser in
+(4) is
+� r
+0 f(x)dx
+� ∞
+r
+xf(x) which is increasing in r. So, if the auto-bidding for budget-constrained bidders is
+not AIC and hence the corresponding ratio is increasing for some r′, it should be increasing for the
+tCPA-constrained advertisers as well, which proves the claim.
+Step 3: Designing a non AIC instance
+The characterization of equilibrium from Step 2 leads us to construct an instance where advertisers
+have the incentive to misreport their constraints. The idea behind the proof is that the value accrued
+by the advertiser 1 is decreasing in r ( as found in Lemma 4.4). Then to find a counter-example, it
+will be enough to find an instance of valuation functions such that the equilibrium equation (3) is
+non-monotone in r.
+Proof of Theorem 4.1. We construct an instance with two budget-constrained advertisers.
+By
+Corollary 4.6 the same instance would work for tCPA-constrained advertisers. To prove the theorem,
+we will find valuation functions v1 and v2 and budgets B1 and B2 such that the value accrued by
+advertiser 1 decreases when their budget increases.
+Define g(r) =
+� r
+0 xf(x)dx
+r
+� ∞
+r
+f(x)dx. By Lemma 4.4, one can find the equilibrium by solving the equation
+g(r) = B2
+B1 . Recall that advertiser 1 wins all queries with v1(q)
+v2(q) ≥ r. So, the total value of queries
+accrued by advertiser 1 is decreasing in r. Hence, to construct a non- AIC example, it is enough to
+find a function f such that g is non-monotone in r.
+A possible such non-monotone function g is
+g(r) = (r − 1)3 + 3
+cr
+− 1,
+(7)
+where c is chosen such that minr≥0 g(r) = 0, i.e., c = min (r−1)3+3
+r
+≈ 1.95105. To see why g is
+non-monotone, observe that g(r) is decreasing for r ≤ 1.8, because g′(r) = 2r3−3r2−2
+cr2
+is negative for
+r ≤ 1.8, and then increasing for r ≥ 1.81.
+We claim the function f defined as in,
+f(r) = 3c(r − 1)2 e
+� r
+0
+c
+(r−1)3+3 dx
+((r − 1)3 + 3)2 ,
+(8)
+would result in the function g in (7). To see why this claim is enough to finish the proof, note that
+there are many ways to choose the value functions of advertisers to derive tf as in (8). One possible
+way is to define v1, v2 : [0, 1] → R as v2(q) = f(tan(q))/(tan(q)2 + 1) and v1(q) = tan(q)v2(q) (see
+Fig. 2).
+12
+
+(a) Valuation function of two advertisers.
+(b) Finding the equilibrium using (3).
+Figure 2: An example of two advertisers such that FPA is not AIC (proof of Theorem 4.1). When
+B1
+B2 = 1200, there are three values for r (see the right panel) that lead to equilibrium, and one
+(orange) leads to non-AIC equilibrium.
+So it remains to prove that choosing f as in (8) would result in g as defined in (7). To derive f
+from g, first we simplify g using integration by part,
+g(r) =
+� r
+0 xf(x)dx
+r
+� ∞
+r
+f(x)dx
+= r
+� r
+0 f(x)dx −
+� r
+0
+� x
+0 f(y)dydx
+r
+� ∞
+r
+f(x)dx
+= r
+� ∞
+0 f(x)dx −
+� r
+0
+� x
+0 f(y)dydx
+r
+� ∞
+r
+f(x)dx
+− 1,
+Assuming that
+� ∞
+0 f(x) is finite, the above equations lead to the following
+rg(r) + r =
+� r
+0
+� ∞
+x f(y)dydx
+� ∞
+r
+f(x)dx
+.
+(9)
+Therefore, by integrating the inverse of both sides,
+log(
+� r
+0
+� ∞
+x
+f(y)dydx) = C +
+� r
+0
+1
+xg(x) + xdx,
+and by raising to the exponent
+� r
+0
+� ∞
+x
+f(y)dydx = Ke
+� r
+0
+1
+xg(x)+x dx.
+for some constants C and K > 0. Then by differentiating both sides with respect to x,
+� ∞
+r
+f(x)dx =
+K
+rg(r) + re
+� r
+0
+1
+xg(x)+x dx.
+Note that for any choice of K ≥ 0, dividing the last two equations will result in (9). So, without loss
+of generality, we can assume K = 1. By differentiating again, we can derive f as a function of g:
+f(r) = (g′(r)r + g(r))
+(rg(r) + r)2 e
+� r
+0
+1
+xg(x)+x dx.
+13
+
+3.0 
+-
+Vi(q)
+25 
+V2(q)
+20 
+15
+LD
+0.5 
+0.D
+0.2
+t0
+0.6
+o.B
+LD
+q3500 
+rE[1[z ≥ r]]
+E[21(z ≤ r)]
+31D0 
+2500
+22000
+1500
+14D0
+50D
+0
+FN
+4
+6
+1
+rWe need g′(r)r + g(r) ≥ 0 to ensure that f(r) ≥ 0 for all r. This holds for g as in (7). Finally, by
+substituting g as in (7), we will derive f as in (8).
+Remark 4.7. Note that the above proof shows that for values of r such that there exists a equilibrium
+which is not AIC, there exists always a second monotone equilibrium. This follows from the fact that
+the function g(r) tends to infinity as r → ∞, so, g must be increasing for some large enough r.
+Before moving on to finding conditions for incentive compatibility, we also note that the above’s
+characterization implies the existence of equilibrium for auto-bidding with any pairs of advertisers.
+Proposition 4.8. Given auto-bidding satisfying the conditions of Lemma 4.4, the equilibrium for
+all pairs of budgets or all pairs of tCPA constrained advertisers always exists.
+Proof. Recall that the equilibirum exists if there exists an r such that
+B2
+B1
+=
+� r
+0 xf(x)dx
+r
+� ∞
+r
+f(x)dx
+has a solution for any value of B2
+B1 . Note that the right-hand side (
+� r
+0 xf(x)dx
+r
+� ∞
+r
+f(x)dx) is positive for any
+r > 0, and it continuously grows to infinity as r → ∞. So, to make sure that every value of
+B2/B1 is covered, we need to check whether at r = 0 the ratio becomes zero. By L’Hopital rule,
+limz→0
+zf(z)
+� ∞
+z
+f(x)dx−zf(z) = 0, which is as desired.
+For tCPA constrained advertiser, the second ratio
+r
+� r
+0 f(x)dx
+� ∞
+r
+xf(x)dx always converges to 0, so the
+equilibrium in this case always exists.
+4.2
+Sufficient Conditions for Incentive Compatibility
+In this section we show that the lack of ACI happens for cases where advertisers’ valuations have
+unusual properties.
+More precisely, the main result of the section is to characterize sufficient
+conditions on the advertiser’s valuations so that FPA is AIC when there are two advertisers in the
+auction.
+For this goal, we recall the function f(z) = v2(h−1(z))
+h′(h−1(z)) where h(q) = v1(q)
+v2(q) defined in Section 4.1.
+As shown in Lemma4.4, function f behaves as a value of the queries advertiser 2 gets and the density
+of queries that advertiser 1 gets.
+Lemma 4.9. Consider that there are two advertisers and they can either both be budget-advertisers
+or tCPA-advertisrs. Also, suppose that auto-bidder with FPA uses the optimal bidding strategy in
+Claim 4.3. Then a sufficient condition for FPA to be AIC is that f has a monotone hazard rate, i.e.,
+f(r)
+� ∞
+r
+f(x)dx is non-decreasing in r.
+Proof. Following the proof Theorem 4.1, if g(r) =
+� r
+0
+� ∞
+x
+f(y)dydx
+r
+� ∞
+r
+f(x)dx
+is non-decreasing in r then the
+equilibrium is AIC. The equivalent sufficient conditions obtained by imposing the inequality g′(r) ≥ 0
+is that for all r ≥ 0,
+r
+� � ∞
+r
+f(x)dx
+�2 ≥
+� � r
+0
+� ∞
+x
+f(y)dydx
+�� � ∞
+r
+f(x)dx − rf(r)
+�
+.
+(10)
+14
+
+If
+� ∞
+r
+f(x)dx ≤ rf(r) then above’s inequality obviously holds. So, we can assume that for some
+r > 0,
+� ∞
+r
+f(x)dx > rf(r). Since
+f(z)
+� ∞
+z
+f(x)dx is non-decreasing in z, we must have that for all r′ ≤ r,
+f(r′)
+� ∞
+r′ f(x)dx ≤
+f(r)
+� ∞
+r
+f(x)dx ≤ 1
+r ≤ 1
+r′ . On the other hand by taking the derivative of
+f(z)
+� ∞
+z
+f(x)dx, we must
+have that f′(z)
+� ∞
+z
+f(x)dx + f(z)2 ≥ 0. By considering two cases on the sign of f′, for z ≤ r we
+must have, f(z)
+�
+zf′(z) + f(z)) ≥ 0, and hence (zf(z))′ ≥ 0 for all z ≤ r. Therefore, zf(z) is
+non-decreasing for z ≤ r.
+On the other hand,
+� r
+0
+� ∞
+x
+f(y)dydx =
+� r
+0
+� r
+x
+f(y)dydx +
+� r
+0
+� ∞
+r
+f(y)dydx
+= r
+� r
+0
+f(x)dx −
+� r
+0
+� x
+0
+f(y)dydx + r
+� ∞
+r
+f(x)dx
+=
+� r
+0
+xf(x)dx + r
+� ∞
+r
+f(x)dx,
+where the second equation is by integration by part. Then by applying monotonicity of z(f)z for
+z ≤ r we have
+� r
+0
+� ∞
+x f(y)dydx ≤ r2f(r) + r
+� ∞
+r
+f(x)dx. So, to prove (10) it is enough to show that
+�� ∞
+r
+f(x)dx
+�2
+≥
+�
+rf(r) +
+� ∞
+r
+f(x)dx
+� �� ∞
+r
+f(x)dx − rf(r)
+�
+,
+which holds, since the right-hand side is equal to
+� � ∞
+r
+f(x)dx
+�2 − (rf(r))2 strictly less than the
+left-hand side.
+While the condition on f has intuitive properties when seen as a density, it has the unappealing
+properties to be too abstract in terms of the conditions on the advertisers’ valuation. The following
+result, provides sufficient conditions on value functions v1 and v2 that makes f be monotone hazard
+rate, and hence, FPA to be AIC.
+Theorem 4.10. Consider two advertisers that are either budget-advertisers or tCPA-advertisers.
+Assume that h(q) = v1(q)
+v2(q) is increasing concave function and that v2 is non-decreasing. Then, the
+equilibrium in FPA auto-bidding with bidding strategy as in Claim 4.3 is AIC.
+Proof. Note that when f is non-decreasing, it also has a monotone hazard rate. Now, when h
+is concave,
+1
+h′ is a non-decreasing function, and since v2 is also non-decreasing, then f is also
+non-decreasing.
+4.3
+FPA with uniform bidding
+The previous section shows that when auto-bidders have full flexibility on the bidding strategy,
+FPA is not AIC. However, non-uniform bid is not simple to implement and auto-bidders may be
+constrained to use simpler uniform bidding policies (aka pacing bidding). In this context, the main
+result of the section is Theorem 4.2 that shows that when restricted to uniform bidding policies FPA
+is AIC. Note that here, we are assuming a simple model where advertisers do not split campaigns.
+So, FPA with uniform bidding is AIC but it could bring up other incentives for advertisers when it
+is implemented.
+15
+
+Definition 4.11 (Uniform bidding equilibrium). A uniform bidding equilibrium for the auto-bidders
+subgame corresponds to bid multipliers µ1, . . . , µN such that every auto-bidder a chooses the uniform
+bidding policy µa that maximizes Problem (1) when restricted to uniform bidding policies with the
+requirement that if advertiser a’s constraints of type 2 are not tight then µa gets its maximum possible
+value.14
+The proof of Theorem 4.2 is based on the main results of Conitzer et al. (2022a). The authors
+proved that the uniform-bidding equilibrium is unique and in equilibrium the multiplier of each
+advertiser is the maximum multiplier over all feasible uniform bidding strategies. Their result is
+for budget-constrained advertisers, and we extend it to include tCPA constrained advertisers. The
+proof is deferred to Appendix B.
+Lemma 4.12 (Extension of Theorem 1 in Conitzer et al. (2022a)). Given an instance of Auto-
+bidding with general constraints as in (2), there is a unique uniform bidding equilibrium, and the bid
+multipliers of all advertisers is maximal among all feasible uniform bidding profiles.
+Now, we are ready to prove Theorem 4.2.
+Proof of Theorem 4.2. Assume that advertiser 1 increases their budget or their target CPA. Then the
+original uniform bidding is still feasible for all advertisers. Further, by Lemma 4.12 the equilibrium
+pacing of all advertisers is maximal among all feasible pacings. So, the pacing of all advertisers
+will either increase or remain the same. But the constraints of all advertisers except 1 are either
+binding or their multiplier has attained its maximum value by the definition of pacing equilibrium.
+Therefore, the set of queries they end up with should be a subset of their original ones since the
+price of all queries will either increase or remain the same. So, it is only advertiser 1 that can win
+more queries.
+Remark 4.13. Conitzer et al. (2022a) show monotonicity properties of budgets in FPA with uniform
+bidding equilibrium for the revenue and welfare. Instead, in our work we focus on monotonicity for
+each advertiser.
+5
+Truthful Auctions
+This section studies auto-bidding incentive compatibility for the case where the per-query auction is
+a truthful auction.
+A truthful auction is an auction where the optimal bidding strategy for a profit-maximizing
+agent is to bid its value. An important example of a truthful auction is Second Price Auction. As we
+showed in the three-queries example of the introduction, SPA is not AIC. In this section, we show
+that the previous example generalizes, in our continuous-query model, to any (randomized) truthful
+auctions so long as the auction is scalar invariant and symmetric (see Assumption 5.1 below for
+details). As part of our proof-technique, we obtain an auction equivalence result which is interesting
+on its own: in the continuous query-model SPA and FPA have the same outcome.15
+For the remaining of the section we assume all truthful auction satisfy the following property.
+14When valuations are strictly positive for all queries q ∈ [0, 1], we can easily show that bid multipliers have to be
+bounded in equilibrium. When this is not the case, we set a cap sufficiently high to avoid bid multipliers going to
+infinity.
+15It is well-known that in the discrete-query model, FPA and SPA are not auction equivalent in the presence of
+auto-bidders.
+16
+
+Assumption 5.1. Let (xa(b))a∈A be the allocation rule in a truthful auction given bids b = (ba)a∈A.
+We assume that the allocation rule satisfies the following properties.
+1. The auction always allocates: �
+a∈A xa(b) = 1
+2. Scalar invariance: For any constant c > 0 and any advertiser a ∈ A, xa(b) = xa(cb).
+3. Symmetry: For any pair of advertisers a, a′ ∈ A and bids b, b′, b−{a,a′} = (b)a∈A\{a,a′} we have
+that
+xa(ba = b, ba′ = b′, b−{a,a′}) = xa′(ba = b′, ba′ = b, b−{a,a′}).
+Remark 5.2. Observe that SPA satisfies Assumption 5.1.
+From the seminal result of Myerson (1981) we obtain a tractable characterization of truthful
+auctions which we use in our proof.
+Lemma 5.3 (Truthful auctions (Myerson, 1981)). Let (xa(b), pa(b))a∈A the allocation and pricing
+rule for an auction given bids b = (ba)a∈A. The auction rule is truthful if and only if
+1. Allocation rule is non-decreasing on the bid: For each bidder a ∈ A and any b′
+a ≥ ba, we have
+that
+xa(b′
+a, b−a) ≥ xa(ba, b−a).
+2. Pricing follows Myerson’s formulae:
+pa(b) = ba · xa(b) −
+� ba
+0
+xa(z, b−a)dz.
+A second appealing property of truthful actions is that the optimal bidding strategy for auto-
+bidders is simpler: in the discrete-query model uniform bidding strategy is almost optimal and can
+differ from optimal by at most the value of two queries (Aggarwal et al., 2019). We revisit this result
+in our continuous-query model and show that uniform bidding policy is optimal for truthful auctions.
+Claim 5.4. In the continuous-query model, if the per-quuery auction is truthful then using a uniform
+bidding is an optimal strategy for each auto-bidder.
+Proof. We use Theorem 1 Aggarwal et al. (2019). Pick some small δ > 0 and divide the interval
+[0, 1] into subintervals of length δ. Let each subinterval I be a discrete query with value functions
+vj(I) =
+�
+I vj(q)dq. Then Theorem 1 Aggarwal et al. (2019) implies that uniform bidding differs from
+optimal by at most two queries. So, the difference from optimal is bounded by 2 maxj max|I|≤δ vj(I).
+Now, since the valuation functions are atomless (i.e., the value of a query is dq), by letting δ to 0,
+the error of uniform bidding in the continuous case also goes to zero.
+5.1
+SPA in the Continuous-Query Model
+We generalize the discrete example of second price auction in Theorem 2.1 to the continuous set
+of queries model showing that SPA is not AIC. The key step consists on showing that for the
+continuous-query model there is an auction equivalence result between first and second price auction.
+17
+
+Theorem 5.5. [Auction Equivalence Result] Suppose that auto-bidder uses a uniform bid strategy
+for SPA, and similarly, uses the simple bidding strategy defined in Claim 4.3 for FPA. Then, in any
+subgame equilibrium the outcome of the auctions (allocations and pricing) on SPA is the same as in
+FPA.
+This result immediately implies that all the results for FPA in Section 4 hold for SPA as well.
+Theorem 5.6. Suppose that there are at least two budget-advertisers or two tCPA-advertisers, then
+even for the continuous-query model SPA is not AIC.
+Similarly to FPA case, we can characterize the equilibrium for the two-advertiser case and derive
+sufficient conditions on advertisers’ valuation functions so that SPA is AIC.
+Theorem 5.7. Given two advertisers, let µ1 and µ2 be the bidding multipliers in equilibrium for the
+subgame of the auto-bidders. Also assume that h(q) = v1(q)
+v2(q) is increasing. Then
+1. If the advertisers are budget-constrained with budget B1 and B2, then µ1 =
+B2
+E[z1(z≥r)] and
+µ2 = µ1r, where r is the answer of the following implicit function,
+rE[1[z ≥ r)]
+E[z1(z ≤ r)] = B1
+B2
+.
+Here, E[.] is defined as E[P(z)] =
+� ∞
+0 P(z)f(z)dz, where f(z) = v2(h−1(z))
+h′(h−1(z)) wherever h′ is
+defined, and it is zero otherwise.
+2. If the advertisers are tCPA-constrained with targets T1 and T2, we have µ1 = T1E[1(z≤r)]
+E[1(z≥r)]
+and
+µ2 = µ1r, where r is the answer of the following implicit function,
+rE[1(z ≥ r)]
+E[z1(z ≥ r)]
+E[1(z ≤ r)]
+E[z1(z ≤ r)] = T1
+T2
+.
+3. If further, v2 is non-decreasing in q, and h is concave, and advertiers are either both budget-
+constrained two tCPA-constrained, then SPA is AIC.
+We now demonstrate the auction equivalence between FPA and SPA.
+Proof of Theorem 5.5. Note that the optimal strategy for a second-price auction is uniform bidding
+with respect to the true value of the query by Claim 5.4. Also, Claim 4.3 implies that the cost obtained
+by each advertiser in first-price auction in the continuous model is also depends on pacing multipliers
+of the other advertiser. This claim immediately, suggests the equivalent between the optimal. bidding
+strategies of first and second price auctions. So, the optimal strategy for both auctions will be the
+same and therefore the resulting allocation and pricing will also be the same. Hence, it follows that
+the same allocation and pricing will be a pure equilibrium under both auctions.
+5.2
+Truthful Auctions Beyond Second-Price
+We now present the main result of the section. We show that a general truthful auction (with
+possibly random allocation) is not AIC.
+18
+
+Theorem 5.8. Consider a truthful auction (x, p) satisfying Assumption 5.1. If there are at least
+two budget-advertisers or two tCPA-advertisers, then the truthful auction is not AIC.
+The remainder of the section gives an overview of the proof of this theorem. Similar to the
+FPA and SPA case, we start by characterizing the equilibrium in the continuous case when there
+are two advertisers in the game. The proof relies on the observation that for auctions satisfying
+Assumption 5.1, the allocation probability is a function of the bids’ ratios. So, again, similar to FPA
+and SPA finding the equilibrium reduces to finding the ratio of bidding multipliers. Then to finish
+the proof of Theorem 5.8 instead of providing an explicit example where auto-bidding is non-AIC, we
+showed that the conditions needed for an auction’s allocation probability to satisfy are impossible.
+The following theorem finds an implicit equation for the best response. We omit the proofs of
+the intermediaries steps and deferred them to the Appendix C.
+Theorem 5.9. Consider a truthful auction (x, p) satisfying Assumption 5.1 and assume that there
+are either two budget-advertisers or two tCPA-advertisers. Let µ1 and µ2 be the bidding multipliers
+used by the auto-bidders in the subgame equilibrium. Further, assume that h(q) = v1(q)
+v2(q) is increasing.
+Then
+1. If the advertisers are budget-constrained with budget B1 and B2, then µ1 =
+B1
+E[p1(rz,1)] and
+µ2 = rµ1, where r is the answer of the following implicit function,
+E[rp1( z
+r, 1)]
+E[zp1( r
+z, 1)] = B1
+B2
+.
+Here, E[.] is defined as E[P(z)] =
+� ∞
+0 P(z)f(z)dz, where f(z) = v2(h−1(z))
+h′(h−1(z)) wherever h′ is
+defined, and it is zero otherwise.
+2. If the advertisers are tCPA-constrained with targets T1 and T2, we have µ1 = T1E[zg(z/r)]
+E[rp1(z/r)] and
+µ2 = µ1r, where r is the answer of the following implicit function,
+E[x1( r
+z, 1)]
+E[zx1( z
+r, 1)]
+E[rp1( z
+r, 1)]
+E[zp1( r
+z, 1)] = T1
+T2
+.
+Because allocation probability x1 is a non-decreasing function, we can derive a similar result to
+the FPA case and show if an instance is not AIC for budget-advertisers then it is also not AIC for
+tCPA-advertisers.
+Proposition 5.10. If for the two budget-constrained advertisers case the truthful auction is not
+AIC, then for the tCPA-constrained advertisers case the same auction is also not AIC.
+Using the previous results we are in position to tackle the main theorem.
+Proof of Theorem 5.8. We prove Theorem 5.8 for budget constrained advertisers, since Proposi-
+tion 5.10 would derive it for tCPA constraint advertisers. We use implicit function theorem to find
+conditions on p1 and f to imply monotonicity in r. Let
+H(x, r) =
+� ∞
+0 rf(z)p1(z/r, 1)dz
+� ∞
+0 f(z)zp1(r/z, 1)dz − x.
+19
+
+Then when advertiser 1 increases budget, the corresponding variable x increases. So, if we want to
+check whether r is a non-decreasing function of x, we need dr
+dx to be non-negative. By the implicit
+function theorem,
+dr
+dx = −
+∂H
+∂x
+∂H
+∂r
+=
+1
+∂H
+∂r
+.
+So, assume to the contrary that r i always non-decreasing in x, then ∂H(x,r
+∂r
+≥ 0.
+Define
+p(x) = p1(x, 1). Then we have the following
+E[ d
+drrp(z/r)]E[zp(r/z)] ≥ E[rp(z/r)]E[ d
+dr
+�
+zp(r/z)
+�
+].
+Then
+d
+drE[rp(z/r)]
+E[rp(z/r)]
+≥
+d
+drE[zp(z/r)]
+E[zp(z/r)]
+By integrating both parts, we have that for any choice of f,
+rE[p(z/r)] ≥ E[zp(r/z)].
+When the above inequality hold for any choice of v1 and v2, we claim that the following must hold
+almost everywhere
+p(b) ≥ bp(1/b).
+(11)
+To see this, assume to the contrary that there exist a measurable set B such that (11) does not hold
+for it. Let qv2(q) = v1(q), therefore, f(z) = v2(z) can be any measurable function. So, we can define
+f to have zero value everywhere except X, and have weight 1 over X to get a contradiction.
+By substituting variable with y = 1/b in (11), p(1/b)db ≥ p(b)/bdb. Therefore, almost everywhere
+p(b) = bp(1/b). By differentiating we have p′(b) = p(1/b) − p′(1/x)/x. On the other hand, as we will
+see in Appendix C for any truthful auction satisfying Assumption 5.1, p′(b) = p′(1/b). Therefore,
+p(b) = p′(b)(b + 1). Solving it for p, we get that the only possible AIC pricing must be of the form
+p(b) = α(b + 1) for some α > 0.
+Next, we will show there is no proper allocation probability satisfying the Assumption 5.1 that
+would result in a pricing function p. It is not hard to see that by the Myerson’s pricing formulae,
+dx1(b,1)
+db
+= p′(b)
+b . Therefore, we must have x′
+1(b, 1) = α/b, so x1(b, 1) = c log(b) + d for some constants
+c > 0 and d. But x1 cannot be a valid allocation rule, since it will take negative values for small
+enough b.
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+https://doi.org/10.1145/3485447.3512062
+Aranyak Mehta and Andres Perlroth. 2023. Auctions without commitment in the auto-bidding world.
+https://doi.org/10.48550/ARXIV.2301.07312
+Roger B. Myerson. 1981. Optimal Auction Design. Mathematics of Operations Research 6, 1 (1981),
+58–73. https://doi.org/10.1287/moor.6.1.58 arXiv:https://doi.org/10.1287/moor.6.1.58
+A
+Second-price tCPA constrained
+Proof. We continue with the proof of Theorem 2.1. We first prove the uniqueness of equilibrium in
+the case of B′
+1 = 1. irst, note that there’s no equilibrium such that advertiser 1 wins all the queries.
+To see this, note that the multiplier of advertiser 1 is at most 1. Hence, the price of q3 for advertiser
+2 is within their budget, and they have the incentive to increase their multiplier to buy q3. Similarly,
+one can see that in any equilibrium, advertiser 1 gets at least q1, since its highest price is within
+their budget.
+Now, assume some equilibrium exists with bidding prices ˜µ1 and ˜µ2 such that advertiser 1 gets
+only q1. Then
+˜µ1(v1(1) + v1(2) + v1(3))
+B2
+> 1 ≥ ˜µ2v2(1)
+B1
+,
+where the first inequality is because advertiser 2’s multiplier is the best response, and the second is
+coming from the budget constraint for advertiser 1. Therefore,
+B1
+B2
+v1(1) + v1(2) + v1(3)
+v2(1)
+≥ ˜µ2
+˜µ1
+,
+But v1(2)
+v2(2) ≥ B1
+B2
+v1(1)+v1(2)+v1(3)
+v2(1)
+= 9
+4, and thus v1(2)
+v2(2) > ˜µ2
+˜µ1 . This is in contradiction with allocation
+inequalities since advertiser 2 wins q2. Therefore, we proved with B1 = 1 and B2 = 4 the equilibrium
+is unique such that advertiser 1 wins q1 and q2.
+22
+
+Now it remains to show a non AIC example for tCPA advertisers. Again consider two advertisers,
+and 3 queries, with the same values as in Table 2. Here, let the tCPA constraint of advertiser 1 be
+T1 = 0.4 and for advertiser 2 be T2 = 0.7. Then again we show that there exists a unique equilibrium
+in which advertiser 1 gets queries 1 and 2.
+First, to prove the existence, let µ1 = 1.6 and µ2 = 1.2. Then we show this is an equilibrium
+since the three following conditoins hold:
+1. Allocation: advertiser 1 wins q1 and q2 since it has a higher bid on them
+v1(1)
+v2(1) ≥ v1(2)
+v2(2) ≥ µ2
+µ1
+= 1.2
+1.5 ≥ v1(3)
+v2(3).
+2. tCPA constraints are satisfied:
+T2v2(3) ≥ µ1v1(3),
+and
+T1(v1(1) + v1(2)) ≥ µ2(v2(1) + v2(2)).
+3. Best response: non of the advertiser can win more queries if they increase their multiplier:
+T2(v2(3) + v2(2)) < µ1(v1(3) + v1(2)),
+and
+T1(v1(1) + v1(2) + v1(3)) < µ2(v2(1) + v2(2) + v2(3)).
+Now, similar to the proof of the budget-constrained advertisers we show the equilibrium is unique.
+Note that there’s no equilibrium such that advertiser 1, gets all queries since the cost of all queries
+for advertiser 1 is at least v2(1) + v2(2) + v2(3) = 12.3 which is larger than T1(v1(1) + v1(2) + v1(3)).
+Similarly, advertiser 2 cannot get all queries since the tCPA constraint would not hold v1(1) +v1(2) +
+v1(3) > T2(v2(1) + v2(2) + v2(3)). So, to prove the uniqueness of equilibrium, it remains to show that
+there’s no equilibrium that advertiser 1 only gets query 1. To contradiction, assume such equilibrium
+exists with the corresponding multipliers ˜µ1 and ˜µ2. Then we must have
+˜µ1(v1(1) + v1(2) + v1(3))
+T2(v2(1) + v2(2) + v2(3)) > 1 ≥ ˜µ2v2(1)
+T1v1(1),
+where the first inequality is because advertiser 2’s multiplier is the best response, and the second
+inequality is coming from the budget constraint for advertiser 1. Therefore,
+T1
+T2
+v1(1) + v1(2) + v1(3)
+v2(1) + v2(2) + v2(3)
+v1(1)
+v2(1) ≥ ˜µ2
+˜µ1
+,
+But v1(2)
+v2(2) ≥ T1
+T2
+v1(1)+v1(2)+v1(3)
+v2(1)+v2(2)+v2(3)
+v1(1)
+v2(1), and thus v1(2)
+v2(2) > ˜µ2
+˜µ1 . This is in contradiction with allocation
+inequalities since advertiser 2 wins q2. Therefore, we proved with T1 = 0.4 and T2 = 0.7, the
+equilibrium is unique such that advertiser 1 wins q1 and q2.
+Now, we show that if advertiser 1 increases their tCPA constraint to T ′
+1 = 0.6, then there exists
+an equilibrium such that advertiser 1 only wins q1. Let µ′
+1 = 1 and µ2 = 2.38. Then
+1. Allocation: advertiser 1 wins q1
+v1(1)
+v2(1) ≥ µ′
+2
+µ′
+1
+= 2.38
+1.
+≥ v1(2)
+v2(2) ≥ v1(3)
+v2(3).
+23
+
+2. tCPA constraints are satisfied:
+T2(v2(3) + v2(2)) ≥ µ′
+1(v1(3) + v1(2)),
+and
+T ′
+1v1(1) ≥ µ′
+2v2(1).
+3. Best response: non of the advertiser can win more queries if they increase their multiplier:
+T2(v2(1) + v2(2) + v2(3)) < µ′
+1(v1(1) + v1(2) + v1(3)),
+and
+T ′
+1(v1(1) + v1(2)) < µ′
+2(v2(1) + v2(2)).
+B
+First-price pacing equilibrium
+Proof of Lemma 4.12. We follow the same steps of the proof as in Conitzer et al. (2022a) for tCPA
+constrained advertisers. Consider two sets of feasible bidding multipliers µ and µ′. We will show
+that µ∗ = max(µ, µ′) is also feasible, where max is the component wise maximum of the bidding
+profiles for n advertisers.
+Each query q is allocated to the bidder with the highest pacing bid. We need to check that
+constraint (2) is satisfied. Fix advertiser a. Its multiplier in µ∗ must be also maximum in one of µ,
+or µ′. without loss assume µ∗
+a = µa. Then the set of queries that a wins with bidding profile µ∗ (X∗
+a)
+must be a subset of queries it wins n µ (Xa), since all other advertisers’ bids have either remained
+the same or increased. On the other hand, the cost of queries a wins stays the same, since it’s a first
+price auction. Since constraint (2) is feasible for bidding multipliers µ we must have
+(µa − Ta)
+�
+q∈X
+va(q) ≤ Ba.
+But then since X∗ ⊆ X, we have as well
+(µa − Ta)
+�
+q∈X∗ va(q) = (µ∗
+a − Ta)
+�
+q∈X∗ va(q) ≤ Ba,
+which implies µ∗ is a feasible strategy.
+To complete the proof we need to show the strategy that all advertisers take the maximum feasible
+pace µ∗
+a = sup{µa|µ is feasible} results in an equilibrium. To see this, note that if an advertiser’s
+strategy is not best-response, they have incentive to increase their pace with its constraints remaining
+satisfied. But then this would result into another feasible pacing strategy and is in contradiction
+with the choice of the highest pace µ∗
+a. A similar argument also shows the equilibrium is unique.
+Assume there exists another pacing equilibrium where an advertiser a exists such that its pace is
+less than µ∗
+a. Then by increasing their pace to µ∗
+a they will get at least as many queries as before, so
+µ∗
+a is the best-response strategy.
+24
+
+C
+Proofs for Truthful Auctions
+We start by the following observation, which follows by applying Assumption 5.1 to reformulate the
+allocation function in the case of two advertisers as a function of a single variable.
+Claim C.1. The probability of allocating each query is a function of the ratio of bids, i.e., there
+exists a non-decreasing function g : R+ → [0, 1] such that the followings hold.16
+1. x1(b1(q), b2(q)) = g( b1(q)
+b2(q)),
+2. g(z) + g(1/z) = 1,
+3. g(0) = 0.
+For example, SPA satisfies the above claim with g(z) = 1 when z = b1(q)
+b2(q) ≥ 1. We are ready to
+prove Theorem 5.9, which follows the similar steps of Lemma 4.4.
+Proof of Theorem 5.9. By Claim 5.4, there exists µ1 and µ2 such that advertiser a bids zava(q) on
+each query. Therefore, we can write the budget constraint for bidder 1 as,
+B1 =
+� 1
+0
+p1(b1(q), b2(q))dq =
+� 1
+0
+µ1v1(q)g
+�v1(q)
+v2(q)
+µ1
+µ2
+�
+dq −
+� 1
+0
+� µ1v1(q)
+0
+g
+�
+x
+v2(q)µ2
+�
+dxdq
+Next, with a change of variable x = v1(q)y we have
+B1 =
+� 1
+0
+µ1v1(q)g
+�v1(q)
+v2(q)
+µ1
+µ2
+�
+dq −
+� 1
+0
+� µ1
+0
+g
+� v1(q)
+v2(q)µ2
+y
+�
+v1(q)dydq.
+As before, let h(q) = v1(q)
+v2(q). Then let z = h(q), we have dq = dh−1(z) =
+1
+h′(h−1(z))dz. So,
+B1 =
+� ∞
+0
+µ1v1(h−1(z))g
+�zµ1
+µ2
+�
+1
+h′(h−1(z))dz −
+� ∞
+0
+� µ1
+0
+g
+� z
+µ2
+y
+�
+v1(h−1(z))dy
+1
+h′(h−1(z))dz.
+Define f(z) = v2(h−1(z))
+h′(h−1(z)) = 1
+z
+v1(h−1(z))
+h′(h−1(z)). Then we have
+B1 =
+� ∞
+0
+µ1zf(z)g
+�zµ1
+µ2
+�
+dz −
+� ∞
+0
+� � µ1
+0
+g
+� z
+µ2
+y
+�
+dy
+�
+zf(z)dz.
+Similarly,
+B2 =
+� ∞
+0
+µ2v2(h−1(z))(1 − g
+�zµ1
+µ2
+�
+)
+1
+h′(h−1(z))dz −
+� ∞
+0
+� µ2
+0
+g
+� y
+µ1z
+�
+v2(h−1(z))dy
+1
+h′(h−1(z))dz.
+B2 =
+� ∞
+0
+µ2f(z)(1 − g
+�zµ1
+µ2
+�
+)dz −
+� ∞
+0
+� µ2
+0
+g
+� y
+µ1z
+�
+dyf(z)dz.
+Next, we find the implicit function to derive r = µ2
+µ1 . By change of variable we have the following
+two equations:
+B1
+µ1
+=
+� ∞
+0
+zf(z)g(z/r)dz − r
+� ∞
+0
+� � z/r
+0
+g(w)dw
+�
+f(z)dz.
+16Notice that the function g is measurable since is non-decreasing.
+25
+
+B2
+µ2
+=
+� ∞
+0
+f(z)(1 − g(z/r))dz − 1
+r
+� ∞
+0
+� � r/z
+0
+g(w)dw
+�
+zf(z)dz.
+The implicit function for r is the following:
+B1
+B2
+=
+� ∞
+0 f(z)
+�
+zg(z/r) − r
+� z/r
+0
+g(w)dw
+�
+dz
+� ∞
+0 f(z)
+�
+r(1 − g(z/r) − z
+� r/z
+0
+g(w)dw
+�
+dz
+.
+Recall the payment rule in Assumption 5.3, this can be re-written as
+B1
+B2
+=
+� ∞
+0 rf(z)p1(z/r, 1)dz
+� ∞
+0 zf(z)zp1(r/z, 1)dz ,
+which finishes the proof for the budget constrained advertisers.
+Now, consider two tCPA constrained advertisers. Following the same argument as above, we get
+the following from tightness of tCPA constraints
+T1
+� ∞
+0
+zf(z)g
+�zµ1
+µ2
+�
+dz =
+� ∞
+0
+µ1zf(z)g
+�zµ1
+µ2
+�
+dz −
+� ∞
+0
+� � µ1
+0
+g
+� z
+µ2
+y
+�
+dy
+�
+zf(z)dz,
+and,
+T2
+� ∞
+0
+f(z)(1 − g
+�zµ1
+µ2
+�
+)dz =
+� ∞
+0
+µ2f(z)(1 − g
+�zµ1
+µ2
+�
+)dz −
+� ∞
+0
+� µ2
+0
+g
+� y
+µ1z
+�
+dyf(z)dz.
+By dividing both sides of the equations we get the desired results.
+Now, to prove the main theorem, we need to show that the values accrued by advertisers is
+monotone in µ1/µ2.
+Claim C.2. Let µi be the optimal bidding multiplier for advertiser i. Given the assumptions in
+Theorem 5.8, the value obtained by advertiser 1 is increasing in r = µ1
+µ2 .
+Proof. Following the proof of Theorem 5.9 we can write value obtained by advertiser i as
+V1(B1, B2) =
+� ∞
+0
+f(z)zg(rz)dz,
+where r is the answer to the implicit function stated in Theorem 5.9. Monotonicity of V1(B1, B2) as
+a function of r follows from the fact that g is a monotone function.
+26
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+arXiv:2301.08618v1  [cs.LG]  20 Jan 2023
+1
+Coupled Physics-informed Neural Networks for
+Inferring Solutions of Partial Differential Equations
+with Unknown Source Terms
+Aina Wang, Pan Qin, Xi-Ming Sun, Senior Member, IEEE,
+Abstract—Physics-informed neural networks (PINNs) provide
+a transformative development for approximating the solutions
+to partial differential equations (PDEs). This work proposes
+a coupled physics-informed neural network (C-PINN) for the
+nonhomogeneous PDEs with unknown dynamical source terms,
+which is used to describe the systems with external forces and
+cannot be well approximated by the existing PINNs. In our
+method, two neural networks, NetU and NetG, are proposed.
+NetU is constructed to generate a quasi-solution satisfying PDEs
+under study. NetG is used to regularize the training of NetU.
+Then, the two networks are integrated into a data-physics-hybrid
+cost function. Finally, we propose a hierarchical training strategy
+to optimize and couple the two networks. The performance of
+C-PINN is proved by approximating several classical PDEs.
+Index Terms—Coupled physics-informed neural network, hier-
+archical training strategy, partial differential equations, unknown
+source term
+I. Introduction
+P
+ARTIAL differential equations (PDEs) are one of the
+general representations for describing spatio-temporal
+dependence in physics [1], medicine [2], engineering [3],
+finance [4], and weather [5], [6]. Numerical approaches, like
+the finite difference method (FDM) [7] and finite element
+(FEM) [8], [9], have been widely investigated and applied.
+FDM used a topologically square lines network to construct
+PDEs’ discretization. Thus, complex geometries in multiple
+dimensions challenge FDM [10]. On the other hand, compli-
+cated geometries can be treated with FEM [11]. The greatest
+difficulty of classical numerical approaches is balancing the
+accuracy and efficiency of forming meshes.
+Among the numerical methods for solving PDEs, the
+Galerkin method is a famous computation method in which the
+linear combination of basis functions was employed to approx-
+imate the solutions to PDEs [12]. Motivated by this, several
+works have used machine learning models to replace the linear
+combination of basis functions to construct data-efficient and
+physics-informed learning methods for solving PDEs [13]–
+[15]. Successful applications of deep learning methods to
+various fields, like image [16], text [17], and speech recogni-
+tion [18], ensure that they are excellent replacers of the linear
+combination of basis functions for solving PDEs [4]. Conse-
+quently, leveraging the well-known approximation capability
+The authors are with the Key Laboratory of Intelligent Control and Opti-
+mization for Industrial Equipment of Ministry of Education and the School
+of Control Science and Engineering, Dalian University of Technology, Dalian
+116024, China e-mail: WangAn@mail.dlut.edu.cn, qp112cn@dlut.edu.cn,
+sunxm@dlut.edu.cn (Corresponding author: Pan Qin)
+of neural networks to solve PDEs is a natural idea and has
+been investigated in various forms previously [19]–[21]. The
+framework of physics-informed neural networks (PINNs) [22]
+was introduced to solve the forward problems while respecting
+any given physical laws governed by PDEs, including the
+nonlinear operator, initial, and boundary conditions. Within
+the PINNs framework, both the sparse measurements and
+the physical knowledge were fully integrated into cost func-
+tion [23], [24]. The solution with respect to spatio-temporal
+dependence was obtained by training the cost function. Note
+that the approximation obtained by machine learning and deep
+learning is meshfree, which has no problem on balancing
+accuracy and efficiency of forming meshes.
+Meanwhile, the potential of using PINNs to solve the inverse
+problem is promising [25]. A hybrid PINN was proposed
+to solve PDEs in [26], in which a local fitting method was
+combined with neural networks to solve PDEs. The hybrid
+PINN was used to identify unknown constant parameters
+in PDEs. The generative adversarial network (GAN) [27]
+was also physics-informed to solve the inverse problems.
+The stochastic physics-informed GAN was investigated for
+estimating the distributions of unknown parameters in PDEs.
+The recent work [28] encoded the physical laws governed
+by PDEs into the architecture of GANs to solve the inverse
+problems for stochastic PDEs. PINNs were also combined with
+the Bayesian method to solve inverse problems from noisy data
+[29].
+PDEs can be classified into homogeneous and nonhomoge-
+neous types. Systems without external forces can be described
+by the homogeneous PDEs. The nonhomogeneous PDEs can
+be applied to reveal the continuous energy propagation be-
+havior of the source and hereby are effective for describing
+practical systems driven by external forces. The function forms
+of the solution and the source term were both assumed to be
+unknown in [30], in which the measurements of the source
+term should be obtained separately from the measurements of
+the solution. However, the independent measurements of the
+external forces cannot always be easily obtained from practical
+situations. The recent work [31] can directly solve the steady-
+state PDEs’ forward and inverse problems, where the source
+terms were assumed to be constant. Thus, [31] was not feasible
+for systems with unsteady external forces, which should be
+described by dynamical functions.
+Although the aforementioned methods have made great
+progress on unknown parameters, prior information or mea-
+surements on external forces cannot always be easily obtained
+
+2
+from practical situations. For example, the real distribution of
+the seismic wave field underground is unknown [32]; the vast
+of signals internal engine, indicating the operation state of
+the engine, cannot be isolated [33]. Furthermore, the existing
+methods with the assumption of the constant source term
+cannot be readily extended to describe the spatio-temporal
+dependence of complex dynamical systems. The determination
+of dynamical source terms with less prior information or even
+without any prior information is an under-investigated issue.
+To this end, this paper proposes a coupled-PINN (C-PINN),
+using the sparse measurements and limited prior information
+of PDEs, to solve PDEs with unknown source terms. In our
+method, two neural networks, NetU and NetG, are proposed.
+NetU is applied to generate a quasi-solution satisfying PDEs
+under study; NetG is used to regularize the training of NetU.
+Then, the two networks are integrated into a data-physics-
+hybrid cost function. Furthermore, we propose a hierarchical
+training strategy to optimize and couple the two networks.
+Finally, the proposed C-PINN is applied to solve several
+classical PDEs to demonstrate its performance.
+The rest of the paper is organized as follows. The classical
+PINNs is briefly reviewed in Section II. A C-PINN using the
+sparse measurements and limited prior knowledge to solve
+PDEs with unknown source terms is proposed in Section III.
+Meanwhile, the two neural networks, NetU and NetG, are
+proposed in our method. Furthermore, a hierarchical training
+strategy is proposed to optimize and couple the two networks.
+In Section IV, our proposed C-PINN is validated with four
+case studies. In Section V, the concluding remarks and the
+future work are presented.
+II. Brief Review of PINNs
+In this section, we briefly review the basic idea of PINNs
+for data-driven solutions to PDEs and data-driven discovery
+of PDEs [22].
+Data-driven solutions to PDEs describe that solve PDEs of
+the generalized form
+ut(x, t) + N[u(x, t)] = 0, x ∈ Ω ⊆ Rd, t ∈ [0, T] ⊂ R
+(1)
+with known parameters. Here, x is the spatial variable, t is
+the temporal variable with t = 0 being at the initial state,
+u : Rd × R → R denotes the hidden solution, N[·] is a
+series of partial differential operators, the domain Ω ⊆ Rd is
+a spatial bounded open set with the boundary ∂Ω. Analytical
+or numerical methods have been widely investigated to find
+proper solution ψ(x, t) satisfying (1) [34]. The left-hand-side
+of (1) can be used to define a residual function as the following
+f(x, t) := ut(x, t) + N[u(x, t)],
+(2)
+where a neural network is used to approximate the solution
+ψ(x, t) to PDEs. The inverse problem is focused on the data-
+driven discovery of PDEs of the generalized form (1), where
+unknown parameters of PDEs here turn into parameters of
+PINNs.
+PINNs for both problems can be trained by minimizing the
+cost function
+MSE = MSED + MSEPH.
+(3)
+Here, MSED is formulated as the following
+MSED =
+�
+(x,t,u)∈D
+�
+ˆu
+�
+x, t; ˆΘU
+�
+− u (x, t)
+�2 ,
+(4)
+where ˆu
+�
+x, t; ˆΘU
+�
+is the function of neural network with ˆΘU
+being its trained parameter set. Let D denote the training
+dataset. This mean squared error term can be considered as
+the data-driven loss. MSEPH is as the following
+MSEPH =
+�
+(x,t)∈E
+ˆf (x, t)2 ,
+(5)
+which regularizes ˆu
+�
+x, t; ˆΘU
+�
+to satisfy (1). Let E denote
+the set of collocation points. This regularization term can be
+considered as the physics-informed loss for the homogeneous
+PDEs. Here, ˆf (x, t) is defined as
+ˆf (x, t) := ˆut
+�
+x, t; ˆΘU
+�
++ N
+�
+ˆu
+�
+x, t; ˆΘU
+��
+,
+(6)
+where ˆut
+�
+x, t; ˆΘU
+�
+and N
+�
+ˆu
+�
+x, t; ˆΘU
+��
+can be obtained using
+automatic differential [35].
+III. Constructing C-PINN
+C-PINN for solving PDEs with unknown source terms is
+presented in this section. The nonhomogeneous PDEs are of
+the following generalized form
+ut(x, t)+N[u(x, t)] = g(x, t), x ∈ Ω ⊆ Rd, t ∈ [0, T] ⊂ R, (7)
+where x and t are the spatial and temporal variable, respec-
+tively, u : Rd ×R → R is similar to (1), g : Rd ×R → R denotes
+the general types of source terms including linear, nonlinear,
+state-steady, or dynamical, Ω is a spatial bounded open set with
+the boundary ∂Ω. Without loss of generality, the spatial set of
+(7) is subjected to Dirichlet boundary, Neumann boundary, or
+the hybrid of Dirichlet and Neumann boundary conditions. In
+general, g(x, t) is used as source terms to describe the external
+forces for dynamical systems and cannot always be separately
+measured, as mentioned in Section I.
+Different from (6), the residual function is defined for the
+nonhomogeneous case as the following
+fN(x, t) := f(x, t)−g(x, t) = ut(x, t)+N[u(x, t)]−g(x, t). (8)
+When g(x, t) is exactly known, ˆfN(x, t), obtained with auto-
+matic differential from (8), can be directly used to regularize
+the approximation of u(x, t). However, the unknown g(x, t)
+will lead to unknown fN(x, t), which makes the aforemen-
+tioned regularization infeasible.
+Therefore, the goal of C-PINN is to approximate the solu-
+tion to PDEs with unknown source terms described by (7). To
+this end, there are two neural networks included in C-PINN:
+(a) NetU for approximating the solution satisfying (7); (b)
+NetG for regularizing the training of NetU.
+1) Cost function:
+To train C-PINN, the training dataset
+is uniformly sampled from the system governed by (7). The
+training dataset D divided into D = DB∪DI with DB∩DI = ∅,
+where DB denotes the boundary and initial training dataset and
+DI is the training dataset of interior of Ω. Collocation points
+
+3
+(x, t) ∈ E correspond to those of (x, t, u) ∈ DI. Then, we adopt
+the following data-physics-hybrid cost function
+MSE = MSED + MSEPN
+(9)
+to train our proposed C-PINN. MSED and MSEPN in (9)
+are the data-driven loss and physics-informed loss for the
+nonhomogeneous PDEs, respectively. MSED adopts the same
+form of (4). MSEPN is as the following
+MSEPN =
+�
+(x,t)∈E
+� ˆf (x, t) − ˆg
+�
+x, t; ˆΘG
+��2 ,
+where ˆg
+�
+x, t; ˆΘG
+�
+is the function of NetG with ˆΘG being
+its trained parameter set,
+ˆf(x, t) has been defined by (2).
+MSEPN corresponds to the physics-informed loss for the
+nonhomogeneous PDEs obtained from (8) imposed at a finite
+set of collocation points (x, t) ∈ E, which is used to regularize
+ˆu
+�
+x, t; ˆΘU
+�
+of NetU to satisfy (7).
+2) Hierarchical training strategy: Considering the relation
+between NetU and NetG in (3), a hierarchical training strategy
+is proposed. In many cases, the exact formulation or even
+sparse measurements of g(x, t) are not available, while the
+sparse measurements DI can be obtained to enforce the
+structure of (7) to achieve ˆΘG. Thus, ΘU and ΘG should be
+iteratively estimated with mutual dependence. Assume k is the
+present iteration step, the core issue of the hierarchical train-
+ing strategy is described by the following two optimization
+problems
+ˆΘ(k+1)
+G
+= arg min
+ΘG
+�
+MSED
+� ˆΘ(k)
+U
+�
++ MSEPN
+�
+ΘG; ˆΘ(k)
+U
+��
+= arg min
+ΘG
+MSEPN
+�
+ΘG; ˆΘ(k)
+U
+�
+(10)
+and
+ˆΘ(k+1)
+U
+= arg min
+ΘU
+�
+MSED (ΘU) + MSEPN
+�
+ΘU; ˆΘ(k+1)
+G
+��
+, (11)
+where ˆΘ(k)
+U is the estimated parameter set of NetU at kth step,
+ˆΘ(k+1)
+G
+is the estimated parameter set of NetG at (k + 1)th step,
+Θ(k+1)
+U
+is the estimated parameter set of NetU at (k + 1)th
+step, which is used to describe the function ˆu
+�
+x, t; ˆΘ(k+1)
+U
+�
+.
+The details of the hierarchical training strategy are obtained
+by Algorithm 1.
+Note that Θ(0)
+U and Θ(0)
+G are used as a given parameter set
+for NetU and the initialization of the parameter set for NetG
+at Step 0, respectively. Furthermore, the iterative transmission
+of parameter sets of NetG and NetU happens in the algorithm.
+IV. Numerical experiments
+In this section, our proposed C-PINN is applied to solve
+several classical PDEs to demonstrate its performance. All
+the examples are implemented with TensorFlow. The fully
+connected structure with a hyperbolic tangent activation func-
+tion is applied, which is initialized by Xavier. These training
+dataset (x, t, u) ∈ D and collocation points (x, t) ∈ E are
+then input into NetU and NetG. L-BFGS [36] is used to
+hierarchically solve the optimization problems (10) and (11)
+to couple the two networks.
+Algorithm 1 The hierarchical strategy of optimizing and
+coupling for C-PINN.
+-Initialize: Randomly sampled training dataset (x, t, u) ∈ D
+and collocation points (x, t) ∈ E. Randomly generate initial
+parameter sets Θ(0)
+U and Θ(0)
+G .
+- Step 0: Assume the kth iteration has achieved ˆΘ(k)
+U and
+ˆΘ(k)
+G .
+Repeat:
+- Step k-1: Training for NetG by solving the optimization
+problem (10) to obtain ˆΘ(k+1)
+G
+, where the estimations of
+ˆut
+�
+x, t; ˆΘ(k)
+U
+�
++ N
+�
+ˆu(x, t; ˆΘ(k)
+U
+�
+in MSEPN is obtained from
+the former iteration result ˆΘ(k)
+U .
+- Step k-2: Training for NetU by solving the optimization
+problem (11) to obtain ˆΘ(k+1)
+U
+, which is used to estimate
+ˆg
+�
+x, t; Θ(k+1)
+G
+�
+in MSEPN.
+-Until the stop criterion is satisfied.
+-Return the solution function ˆΘU → ˆu
+�
+x, t; ˆΘU
+�
+, which can
+predict the solution (8) with any point (x, t) in Ω.
+We evaluate the performance of our proposed C-PINN by
+means of root mean squared error (RMSE)
+RMSE =
+�
+1
+|T|
+�
+(x,t)∈T
+(u (x, t) − ˆu (x, t))2,
+where |T| is the cardinality with respect to the collocation
+points (x, t) ∈ T, T is the set of testing collocation points.
+u (x, t) and ˆu (x, t) denote the ground truth and the cor-
+responding predictions, respectively. To further validate the
+performance of C-PINN, the Pearson’s correlation coefficient
+(CC)
+CC =
+cov (u (x, t) , ˆu (x, t))
+√Var u (x, t) √Var ˆu (x, t)
+is also used to measure the similarity between ground truth and
+prediction, where CC is the correlation coefficient of u(x, t)
+and ˆu(x, t), cov (u (x, t) , ˆu (x, t)) is the covariance between
+u(x, t) and ˆu(x, t), and Var u (x, t) and Var ˆu (x, t) are variance
+of u(x, t) and ˆu(x, t), respectively.
+A. Case 1: 1-D Heat Equation
+C-PINN is first applied to solve the heat equation with
+unknown external forces, where both Dirichlet and Neumann
+boundary conditions are conducted to demonstrate its perfor-
+mance.
+1) Dirichlet Boundary Condition
+Here, we consider the heat equation with Dirichlet boundary
+condition as the following
+∂u
+∂t = a2 ∂2u
+∂x2 + g(x, t),
+0 < x < L, t > 0
+u|t=0 = φ(x),
+0 ⩽ x ⩽ L
+u|x=0 = 0,
+u|x=L = 0,
+t > 0,
+(12)
+where thermal diffusivity a
+=
+1, u(x, t) is the primary
+variable and means the temperature at (x, t), L = π is the
+length of bounded rod, φ(x) = 0 is initial temperature, and
+
+4
+g(x, t) = xe−t denotes the unknown external heat source at
+(x, t). The analytical solution u (x, t) to (12) is obtained with
+respect to [37]. In this experiment, the setting-ups of C-PINN
+are as follows. There are eight hidden layers with 20 units
+in each of them for both NetU and NetG. A total of 110
+training data (x, t, u(x, t)) in D with t ∈ [0, 6], including 10
+training data in DI and 100 training data in DB, are randomly
+sampled, 10 sparse collocation points are randomly sampled to
+enforce the structure of (12). Fig. 1 shows the sparse training
+dataset and the prediction results. Specifically, the magnitude
+of the predictions ˆu(x, t) using the training dataset is shown in
+Fig. 1(a) with a heat map. In this case, RMSE is 4.225390e−02
+and the correlation coefficient is 9.785444e − 01. Moreover,
+we compare the ground truths and the predictions at fixed-
+time t= 1.5, 3, and 4.5 in Fig. 1(b) to (d), respectively. The
+evaluation criteria in Table I are applied to further quantify
+the performance of our proposed C-PINN.
+0
+1
+2
+3
+4
+5
+6
+t
+0
+1
+2
+3
+x
+��� ������������������uˆ(x, t)
+����training data �����
+���training data ������
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+2
+x
+0
+1
+u(x, t)
+��� t = 1.5
+0
+2
+x
+0
+1
+u(x, t)
+��� t = 3
+0
+2
+x
+0
+1
+u(x, t)
+��� t = 4.5
+������������
+����������
+Fig. 1.
+(a) Predictions ˆu (x, t) for the 1-D heat equation with Dirichlet
+boundary condition; (b), (c), and (d) Comparisons of the ground truths and
+predictions corresponding to the fixed-time t= 1.5, 3, and 4.5 snapshots
+depicted by the dashed vertical lines in (a), respectively.
+TABLE I
+Evaluation criteria for the three temporal snapshots depicted by the dashed
+vertical lines in Fig. 1-(a)
+Criteria
+1.5
+3
+4.5
+RMSE
+4.600305e-02 1.342719e-02
+2.991229e-02
+CC
+9.753408e-01 9.912983e-01
+9.805664e-01
+Subsequently, the experiment for PDE with Neumann
+boundary condition will be further explored to show the
+general performance of C-PINN.
+2) Neumann Boundary Condition
+Heat equation with Neumann boundary condition is defined
+as
+∂u
+∂t = a2 ∂2u
+∂x2 + g(x, t),
+0 < x < L, t > 0
+u|t=0 = φ(x),
+0 ⩽ x ⩽ L
+u|x=0 = 0,
+∂u
+∂x
+�����x=L
+= 0,
+t > 0,
+(13)
+with the thermal diffusivity a = 1, the length of bounded
+rod L = π, the initial temperature φ(x) = sin (x/2), and
+the external heat source is g(x, t) = sin (x/2). The analytical
+solution u(x, t) to (13) is obtained according to [37]. In this
+example, NetU is of three hidden layers consisting of 30
+neurons individually. NetG is of eight hidden layers consisting
+of 20 units individually. (x, t, u(x, t)) in D are considered with
+t ∈ [0, 10]. A total of 130 training data in DB, including 10
+initial training data, 60 left boundary training data, and 60 right
+boundary training data are randomly sampled. Moreover, the
+20 sparse collocation points are randomly sampled to enforce
+the structure of (13). The magnitude of the predictions ˆu(x, t)
+using the training dataset is shown in Fig. 2(a). RMSE is
+5.748950e−02 and the correlation coefficient is 9.988286e−01.
+Moreover, we compare the ground truths and the predictions at
+fixed-time t= 3, 6, and 9 in Fig. 2(b) to (d), respectively. The
+evaluation criteria in Table II are applied to further evaluate
+the performance of our proposed C-PINN.
+0
+2
+4
+6
+8
+10
+t
+0
+1
+2
+3
+x
+��� ������������������ uˆ(x, t)
+130 training data������
+20 training data ������
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0
+2
+x
+0
+1
+2
+3
+u(x, t)
+��� t = 3
+0
+2
+x
+0
+1
+2
+3
+u(x, t)
+��� t = 6
+0
+2
+x
+0
+1
+2
+3
+u(x, t)
+��� t = 9
+�������������
+Prediction
+Fig. 2.
+(a) Predictions ˆu (x, t) for the 1-D heat equation with Neumann
+boundary condition. (b), (c), and (d) Comparisons of the ground truths and
+predictions correspond to the fixed-time t= 3, 6, and 9 snapshots depicted by
+the dashed vertical lines in (a), respectively.
+TABLE II
+Evaluation criteria for the three temporal snapshots depicted by the dashed
+vertical lines in Fig. 2-(a).
+Criteria
+3
+6
+9
+RMSE
+5.343142e-02
+5.884118e-02
+7.064205e-02
+CC
+9.982448e-01
+9.990231e-01
+9.984719e-01
+B. Case 2: 1-D Wave Equation
+The wave equation is as the following
+∂2u
+∂t2 = a2 ∂2u
+∂x2 + g(x, t),
+0 < x < L, t > 0
+u|x=0 = 0,
+u|x=L = 0,
+t > 0
+u|t=0 = 0,
+∂u
+∂t
+�����t=0
+= 0,
+0 ⩽ x ⩽ L,
+(14)
+
+5
+where the wave speed a is 1, the length of bounded string L
+is π, the time of wave propagation t is 6, the external force is
+g(x, t) = sin 2πx
+L sin 2aπt
+L
+at(x, t) and displacement u(x, t) at (x, t) according to [37] is
+further investigated.
+In this experiment, NetU is of three hidden layers consisting
+of 30 neurons individually. NetG is of eight hidden layers
+consisting of 20 units individually. A total of 210 training
+data (x, t, u (x, t)) in D, including 50 initial training data, 120
+boundary training data, and 40 collation points are randomly
+sampled. Fig. 3(a) shows the sparse training dataset and the
+magnitude of displacement ˆu(x, t) at (x, t). Fig. 3(b) to (d)
+show the comparisons of ground truths and predictions corre-
+sponding to the three fixed-time t=1.5, 3, and 4.5, which are
+depicted by the dashed vertical lines in Fig. 3(a), respectively.
+RMSE is 7.068626e − 02 and the correlation coefficient is
+9.864411e− 01. The evaluation criteria for the three temporal
+snapshots are listed in Table III.
+0
+1
+2
+3
+4
+5
+6
+�
+0
+1
+2
+3
+x
+(�� ����������������� ˆu(x, t)
+220 training data �����
+40 training data �����
+−1.0
+−0.5
+0.0
+0.5
+1.0
+0
+2
+x
+−1
+0
+1
+u(x, t)
+�b� �������
+0
+2
+x
+−1
+0
+1
+u(x, t)
+��� �����
+0
+2
+x
+−1
+0
+1
+u(x, t)
+��� �������
+�������������
+����������
+Fig. 3.
+(a) Predictions ˆu (x, t) for 1-D wave equation. (b), (c), and (d)
+Comparisons of the ground truths and predictions corresponding to the fixed-
+time t=1.5, 3, and 4.5 snapshots depicted by the dashed vertical lines in (a),
+respectively.
+TABLE III
+Evaluation criteria for the three temporal snapshots depicted by the dashed
+vertical lines in Fig. 3-(a).
+Criteria
+1.5
+3.
+4.5
+RMSE
+1.424030e-01
+3.305190e-02
+5.201132e-02
+CC
+9.6238994e-01
+9.985312e-01
+9.983170e-01
+C. Case 3: 2-D Poisson Equation
+We further consider the following 2-D Poisson equation
+∂2u
+∂x2 + ∂2u
+∂y2 = T0,
+0 < x < a, 0 < y < b
+u(x, 0) = 0,
+u(x, b) = T,
+0 ⩽ x ⩽ a
+u(0, y) = 0,
+u(a, y) = 0,
+0 ⩽ y ⩽ b,
+(15)
+where T is 1, the constant source term T0 = 1 is unknown, and
+a = b = 1. The analytical solution u(x, y) to (15) is obtained
+according to [37]. In this experiment, the setting-ups of C-
+PINN are as follows. There are eight hidden layers with 20
+units in each of them for both NetU and NetG. Thirty training
+data in DB and 3 collocation points in DI are used. Fig. 4(a)
+shows the sparse training dataset and the predictions ˆu(x, y).
+Fig. 4(b) to (d) show the prediction performance of fixed-
+location y=0.2, 0.4, and 0.6 snapshots depicted in Fig. 4(a),
+respectively. RMSE is 1.594000e − 02 and the correlation
+coefficient is 9.997390e − 01. The corresponding evaluation
+criteria are listed in Table IV.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+x
+��� ���������������������uˆ(x, y)
+30 training data ������
+3 training data �����
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0
+1
+x
+0.0
+0.5
+1.0
+1.5
+u(x, y)
+0
+1
+x
+0.0
+0.5
+1.0
+1.5
+u(x, y)
+0
+1
+x
+0.0
+0.5
+1.0
+1.5
+u(x, y)
+(���   �����
+�������������
+����������
+y
+y
+(c��   ����4
+y
+(b��   ����2
+y
+Fig. 4.
+(a) Predictions ˆu (x, y) for the 2-D Poisson equation. (b), (c), and,
+(d) Comparisons of the ground truths and predictions corresponding to the
+fixed-location y = 0.2, 0.4, and 0.6 snapshots depicted by the dashed vertical
+lines in (a), respectively.
+TABLE IV
+Evaluation criteria for the three fixed-location snapshots depicted by the
+dashed vertical lines in Fig. 4-(a).
+Criteria
+0.2
+0.4
+0.6
+RMSE
+1.763408e-02
+1.139888e-02
+7.696680e-03
+CC
+9.986055e-01
+9.999703e-01
+9.999656e-01
+D. Case 4: 3-D Helmholtz Equation
+C-PINN is also applied to solve 3-D Helmholtz equation
+with an unknown source term. In particular, we consider the
+same test PDEs that were previously suggested in [26]
+∆u(x) + p2u(x) = g(x) in Ω ⊂ R3
+u(x) = u0(x) on ∂Ω,
+(16)
+where ∆ =
+∂
+∂x2 + ∂
+∂y2 + ∂
+∂z2 is Laplacian operator, x = (x, y, z)⊤
+is coordinates with x, y, z ∈ (0, 1/4] , p = 5 is the wavenumber,
+
+6
+a suitable g(x) is the right-hand side of (16) so that
+u(x) = (0.1 sin (2πx) + tanh (10x)) sin (2πy)sin (2πz)
+is the analytical solution of (16) [26]. In this experiment,
+NetU is of three hidden layers consisting of 100, 50, and
+50 neurons individually. NetG is of eight hidden layers con-
+sisting of 20 units individually. Sixty training data and 120
+collocation points are sampled. Fig. 5(a) shows the solution
+of (x, y, z = 0.12) snapshot. Furthermore, Fig. 5(b) to (d)
+show the comparisons of ground truths and predictions, which
+are extracted at (x = 0.05, z = 0.12), (x = 0.15, z = 0.12),
+and (x = 0.2, z = 0.12), respectively. The evaluation criteria
+for this extractions are listed in Table V. In this experiment,
+RMSE is 1.192859e − 01, and the correlation coefficient is
+9.057524e − 01.
+0.05
+0.10
+0.15
+0.20
+0.25
+x
+0.1
+0.2
+y
+(a) 3−D Helmholtz Equation             
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.10.2
+y
+0.0
+0.2
+0.4
+0.6
+u(x, y)
+�b� x = 0.05
+0.10.2
+y
+0.0
+0.2
+0.4
+0.6
+u(x, y)
+��� x = 0.15
+0.10.2
+y
+0.0
+0.2
+0.4
+0.6
+u(x, y)
+��� x =0.2
+�������������
+Prediction
+�uˆ(x,�y,�z�=�25
+3�)
+Fig. 5. (a) Predictions ˆu (x, y, z = 0.12) for 3-D Helmholtz equation. (b), (c)
+and, (d) Comparisons of the ground truths and predictions corresponding to
+the (x = 0.05, z = 0.12), (x = 0.15, z = 0.12), and (x = 0.20, z = 0.12)
+snapshots depicted by the dashed vertical lines in (a), respectively.
+TABLE V
+Evaluation criteria for the three snapshots depicted by the dashed vertical
+lines in Fig. 5-(a).
+Criteria
+0.05
+0.15
+0.2
+RMSE
+7.043735e-02
+7.548533e-02
+5.179414e-02
+CC
+9.604538e-01
+9.998589e-01
+9.964517e-01
+V. Conclusion
+This paper proposes a novel PINN, called C-PINN, to solve
+PDEs with less prior information or even without any prior
+information for source terms. In our approach, two neural net-
+works, NetU and NetG, are proposed with a fully-connected
+structure. NetU for approximating the solution satisfying
+PDEs under study; NetG for regularizing the training of NetU.
+Then, the two networks are integrated into a data-physics-
+hybrid cost function. Furthermore, the two networks are op-
+timized and coupled by the proposed hierarchical training
+strategy. Finally, C-PINN is applied to solve several classical
+PDEs to testify to its performance. Note that C-PINN inherits
+the advantages of PINN, such as sparse property and automatic
+differential. C-PINN is proposed to solve such a dilemma as
+the governing equation of dynamical systems with unknown
+forces. Thus, C-PINN can be further applied to infer the
+unknown source terms. Meanwhile, C-PINN can be extended
+to identify the operators from the sparse measurements.
+In the future, we will continue to use our C-PINN in
+various scenarios, like solving PDEs with unknown struc-
+ture parameters and high-dimension PDEs. For the case, the
+structures of PDE are totally unknown, regularization method
+will be combined with C-PINN to select operators from
+the sparse measurements. Our proposed C-PINN has been
+shown to solve several classical PDEs successfully. For more
+complex situations, the features extraction, like convolution
+and pooling, will be added to C-PINN.
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+

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+arXiv:2301.13789v1  [math.CO]  31 Jan 2023
+The Minimum Degree Removal Lemma Thresholds
+Lior Gishboliner∗
+Zhihan Jin∗
+Benny Sudakov∗
+Abstract
+The graph removal lemma is a fundamental result in extremal graph theory which says that for every
+fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H then G contains
+δnv(H) copies of H for some δ = δ(ε, H) > 0. The current proofs of the removal lemma give only very
+weak bounds on δ(ε, H), and it is also known that δ(ε, H) is not polynomial in ε unless H is bipartite.
+Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ(ε, H)
+depends polynomially or linearly on ε. In this paper we answer several questions of Fox and Wigderson
+on this topic.
+1
+Introduction
+The graph removal lemma, first proved by Ruzsa and Szemerédi [23], is a fundamental result in extremal
+graph theory. It also have important applications to additive combinatorics and property testing. The lemma
+states that for every fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H
+then G it contains δnv(H) copies of H, where δ = δ(ε, H) > 0. Unfortunately, the current proofs of the graph
+removal lemma give only very weak bounds on δ = δ(ε, H) and it is a very important problem to understand
+the dependence of δ on ε. The best known result, due to Fox [11], proves that 1/δ is at most a tower of
+exponents of height logarithmic in 1/ε. Ideally, one would like to have better bounds on 1/δ, where an
+optimal bound would be that δ is polynomial in ε. However, it is known [2] that δ(ε, H) is only polynomial
+in ε if H is bipartite. This situation led Fox and Wigderson [12] to initiate the study of minimum degree
+conditions which guarantee that δ(ε, H) depends polynomially or linearly on ε. Formally, let δ(ε, H; γ) be
+the maximum δ ∈ [0, 1] such that if G is an n-vertex graph with minimum degree at least γn and with εn2
+edge-disjoint copies of H, then G contains δnv(H) copies of H.
+Definition 1.1. Let H be a graph.
+1. The linear removal threshold of H, denoted δlin-rem(H), is the infimum γ such that δ(ε, H; γ) depends
+linearly on ε, i.e. δ(ε, H; γ) ≥ µε for some µ = µ(γ) > 0 and all ε > 0.
+2. The polynomial removal threshold of H, denoted δpoly-rem(H), is the infimum γ such that δ(ε, H; γ)
+depends polynomially on ε, i.e. δ(ε, H; γ) ≥ µε1/µ for some µ = µ(γ) > 0 and all ε > 0.
+Trivially, δlin-rem(H) ≥ δpoly-rem(H).
+Fox and Wigderson [12] initiated the study of δlin-rem(H) and
+δpoly-rem(H), and proved that δlin-rem(Kr) = δpoly-rem(Kr) = 2r−5
+2r−3 for every r ≥ 3, where Kr is the clique on
+r vertices. They further asked to determine the removal lemma thresholds of odd cycles. Here we completely
+resolve this question. The following theorem handles the polynomial removal threshold.
+Theorem 1.2. δpoly-rem(C2k+1) =
+1
+2k+1.
+Theorem 1.2 also answers another question of Fox and Wigderson [12], of whether δlin-rem(H) and
+δpoly-rem(H) can only obtain finitely many values on r-chromatic graphs H for a given r ≥ 3. Theorem 1.2
+shows that δpoly-rem(H) obtains infinitely many values for 3-chromatic graphs. In contrast, δlin-rem(H) ob-
+tains only three possible values for 3-chromatic graphs. Indeed, the following theorem determines δlin-rem(H)
+for every 3-chromatic H. An edge xy of H is called critical if χ(H − xy) < χ(H).
+∗Department of Mathematics, ETH, Zürich, Switzerland. Research supported in part by SNSF grant 200021_196965. Email:
+{lior.gishboliner, zhihan.jin, benjamin.sudakov}@math.ethz.ch.
+1
+
+Theorem 1.3. For a graph H with χ(H) = 3, it holds that
+δlin-rem(H) =
+
+
+
+
+
+1
+2
+H has no critical edge,
+1
+3
+H has a critical edge and contains a triangle,
+1
+4
+H has a critical edge and odd-girth(H) ≥ 5.
+Theorems 1.2 and 1.3 show a separation between the polynomial and linear removal thresholds, giving a
+sequence of graphs (i.e. C5, C7, . . . ) where the polynomial threshold tends to 0 while the linear threshold is
+constant 1
+4.
+The parameters δpoly-rem and δlin-rem are related to two other well-studied minimum degree thresholds:
+the chromatic threshold and the homomorphism threshold. The chromatic threshold of a graph H is the
+infimum γ such that every n-vertex H-free graph G with δ(G) ≥ γn has bounded cromatic number, i.e.,
+there exists C = C(γ) such that χ(G) ≤ C. The study of the chromatic threshold originates in the work of
+Erdős and Simonovits [10] from the ’70s. Following multiple works [4, 15, 16, 7, 5, 25, 26, 19, 6, 14, 20], the
+chromatic threshold of every graph was determined by Allen et al. [1].
+Moving on to the homomorphism threshold, we define it more generally for families of graphs. The
+homomorphism threshold of a graph-family H, denoted δhom(H), is the infimum γ for which there exists an
+H-free graph F = F(γ) such that every n-vertex H-free graph G with δ(G) ≥ γn is homomorphic to F.
+When H = {H}, we write δhom(H). This parameter was widely studied in recent years [18, 22, 17, 8, 24].
+It turns out that δhom is closely related to δpoly-rem(H), as the following theorem shows. For a graph H, let
+IH denote the set of all minimal (with respect to inclusion) graphs H′ such that H is homomorphic to H′.
+Theorem 1.4. For every graph H, δpoly-rem(H) ≤ δhom(IH).
+Note that IC2k+1 = {C3, . . . , C2k+1}. Using this, the upper bound in Theorem 1.2 follows immediately
+by combining Theorem 1.4 with the result of Ebsen and Schacht [8] that δhom({C3, . . . , C2k+1}) =
+1
+2k+1.
+The lower bound in Theorem 1.2 was established in [12]; for completeness, we sketch the proof in Section 3.
+The rest of this short paper is organized as follows. Section 2 contains some preliminary lemmas. In
+Section 3 we prove the lower bounds in Theorems 1.2 and 1.3. Section 4 gives the proof of Theorem 1.4, and
+Section 5 gives the proof of the upper bounds in Theorem 1.3. In the last section we discuss further related
+problems.
+2
+Preliminaries
+Throughout this paper, we always consider labeled copies of some fixed graph H and write copy of H for
+simplicity. We use δ(G) for the minimum degree of G, and write H → F to denote that there is a homo-
+morphism from H to F. For a graph H on [h] and integers s1, s2, . . . , sh > 0, we denote by H[s1, . . . , sh]
+the blow-up of H where each vertex i ∈ V (H) is replaced by a set Si of size si (and edges are replaced with
+complete bipartite graphs). The following lemma is standard.
+Lemma 2.1. Let H be a fixed graph on vertex set [h] and let s1, s2, . . . , sh ∈ N. There exists a constant
+c = c(H, s1, . . . , sh) > 0 such that the following holds. Let G be an n-vertex graph and V1, . . . , Vh ⊆ V (G).
+Suppose that G contains at least ρnh copies of H mapping i to Vi for all i ∈ [h]. Then G contains at least
+cρ
+1
+c · ns1+···+sh copies of H[s1, . . . , sh] mapping Si to Vi for all i ∈ [h].
+Note that the sets V1, . . . , Vh in Lemma 2.1 do not have to be disjoint. The proof of Lemma 2.1 works
+by defining an auxiliary h-uniform hypergraph G whose hyperedges correspond to the copies of H in which
+vertex i is mapped to Vi. By assumption, G has at least ρnh edges. By the hypergraph generalization of the
+Koväri-Sós-Turán theorem, see [9], G contains poly(ρ)ns1+···+sh copies of K(h)
+s1,...,sh, the complete h-partite
+hypergraph with parts of size s1, . . . , sh. Each copy of K(h)
+s1,...,sh gives a copy of H[s1, . . . , sh] mapping Si to Vi.
+Fox and Wigderson [12, Proposition 4.1] proved the following useful fact.
+Lemma 2.2. If H → F and F is a subgraph of H, then δpoly-rem(H) = δpoly-rem(F).
+2
+
+The following lemma is an asymmetric removal-type statement for odd cycles, which gives polynomial
+bounds. It may be of independent interest. A similar result has appeared very recently in [13].
+Lemma 2.3. For 1 ≤ ℓ < k, there exists a constant c = c(k) > 0 such that if an n-vertex graph G has εn2
+edge-disjoint copies of C2ℓ+1, then it has at least cε1/cn2k+1 copies of C2k+1.
+Proof. Let C be a collection of εn2 edge-disjoint copies of C2ℓ+1 in G. There exists a collection C′ ⊆ C such
+that |C′| ≥ εn2/2 and each vertex v ∈ V (G) belongs to either 0 or at least εn/2 of the cycles in C′. Indeed,
+to obtain C′, we repeatedly delete from C all cycles containing a vertex v which belongs to at least one but
+less than εn/2 of the cycles in C (without changing the graph). The set of cycles left at the end is C′. In
+this process, we delete at most εn2/2 cycles altogether (because the process lasts for at most n steps); hence
+|C′| ≥ εn2/2. Let V be the set of vertices contained in at least εn/2 cycles from C′, so |V | ≥ εn/2. With
+a slight abuse of notation, we may replace G with G[V ], C with C′ and ε/2 with ε, and denote |V | by n.
+Hence, from now on, we assume that each vertex v ∈ V (G) is contained in at least εn of the cycles in C.
+This implies that |N(v)| ≥ 2εn for every v ∈ V (G).
+Fix any v0 ∈ V (G) and let C(v0) be the set of cycles C ∈ C such that C ∩ N(v0) ̸= ∅ and v0 /∈ C.
+The number of cycles C ∈ C intersecting N(v0) is at least |N(v0)| · εn/(2ℓ + 1) ≥ 2ε2n2/(2ℓ + 1), and the
+number of cycles containing v0 is at most n. Hence, |C(v0)| ≥ 2ε2n2/(2ℓ + 1) − n ≥ ε2n2/(ℓ + 1). Take
+a random partition V0, V1, . . . , Vℓ of V (G) \ {v0}, where each vertex is put in one of the parts uniformly
+and independently. For a cycle (x1, . . . , x2ℓ+1) ∈ C(v0) with xℓ+1 ∈ N(v0), say that (x1, . . . , x2ℓ+1) is good
+if xℓ+1 ∈ V0 and xℓ+1−i, xℓ+1+i ∈ Vi for 1 ≤ i ≤ ℓ (so in particular x1, x2ℓ+1 ∈ Vℓ).
+The probability
+that (x1, . . . , x2ℓ+1) is good is 1/(ℓ + 1)2ℓ+1, so there is a collection of good cycles C′(v0) ⊆ C0 of size
+|C′(v0)| ≥ |C(v0)|/(ℓ + 1)2ℓ+1 ≥ ε2n2/(ℓ + 1)2ℓ+2.
+Put γ := ε2/(ℓ + 1)2ℓ+2.
+By the same argument as
+above, there is a collection C′′(v0) ⊆ C′(v0) with |C��′(v0)| ≥ γn2/2 such that each vertex is contained in
+either 0 or at least γn/2 cycles from C′′(v0). Let W be the set of vertices contained in at least γn/2 cycles
+from C′′(v0). Note that W ∩ V0 ⊆ N(v0) by definition. Also, each vertex in W ∩ Vℓ has at least γn/2
+neighbors in W ∩ Vℓ, and for each 1 ≤ i ≤ ℓ, each vertex in W ∩ Vi has at least γn/2 neighbors in W ∩ Vi−1.
+It follows that W ∩ Vℓ contains at least 1
+2|W ∩ Vℓ| · �2k−2ℓ−2
+i=0
+(γn/2 − i) = poly(γ)n2k−2ℓ paths of length
+2k − 2ℓ − 1. We now construct a collection of copies of C2k+1 as follows. Choose a path yℓ+1, yℓ+2, . . . , y2k−ℓ
+of length 2k − 2ℓ − 1 in W ∩ Vℓ.
+For each i = ℓ, . . . , 1, take a neighbor yi ∈ W ∩ Vi−1 of yi+1 and a
+neighbor y2k−i+1 ∈ W ∩ Vi−1 of y2k−i, such that the vertices y1, . . . , y2k are all different. Then y1, . . . , y2k
+is a path and y1, y2k ∈ W ∩ V0 ⊆ N(v0), so v0, y1, . . . , y2k is a copy of C2ℓ+1. The number of choices for
+the path yℓ+1, yℓ+2, . . . , y2k−ℓ is poly(γ)n2k−2ℓ and the number of choices for each vertex yi, y2k−i+1 ∈ Vi−1
+(i = ℓ, . . . , 1) is at least γn/2. Hence, the total number of choices for y1, . . . , y2k is poly(γ)n2k. As there are
+n choices for v0, we get a total of poly(γ)n2k+1 = polyk(ε)n2k+1 copies of C2k+1, as required.
+3
+Lower bounds
+Here we prove the lower bounds in Theorems 1.2 and 1.3. The lower bound in Theorem 1.2 was proved in
+[12, Theorem 4.3]. For completeness, we include a sketch of the proof:
+Lemma 3.1. δpoly-rem(C2k+1) ≥
+1
+2k+1.
+Proof. Fix an arbitrary α > 0. In [2] it was proved that for every ε, there exists a (2k + 1)-partite graph
+with parts V1, . . . , V2k+1 of size αn/(2k + 1) each, with εn2 edge-disjoint copies of C2k+1, but with only
+εω(1)n2k+1 copies of C2k+1 in total (where the ω(1) term may depend on α). Add sets U1, . . . , U2k+1 of size
+(1 − α)n/(2k + 1) each, and add the complete bipartite graphs (Ui, Vi), 1 ≤ i ≤ 2k + 1, and (Ui, Ui+1),
+1 ≤ i ≤ 2k. See Figure 1. It is easy to see that this graph has minimum degree (1 − α)n/(2k + 1), and every
+copy of C2k+1 is contained in V1 ∪ · · · ∪ V2k+1. Letting α → 0, we get that δpoly-rem(C2k+1) ≥
+1
+2k+1.
+By combining the fact that δpoly-rem(C3) = 1
+3 with Lemma 2.2 (with F = C3), we get that δlin-rem(H) ≥
+δpoly-rem(H) = 1
+3 for every 3-chromatic graph H containing a triangle. This proves the lower bound in the
+second case of Theorem 1.3. Now we prove the lower bounds in the other two cases. We prove a more general
+statement for r-chromatic graphs.
+3
+
+V2
+V3
+V4
+V5
+V1
+U2
+U3
+U4
+U5
+U1
+Figure 1: Proof of Lemma 3.1 for C5. Heavy edges indicate complete bipartite graphs while dashed edges
+form the Ruzsa–Szemerédi construction for C5 (see [2]).
+Lemma 3.2. Let H be a graph with χ(H) = r ≥ 3.
+Then,
+3r−8
+3r−5 ≤ δlin-rem(H) ≤
+r−2
+r−1.
+Moreover,
+δlin-rem(H) = r−2
+r−1 if H contains no critical edge.
+Proof. Denote h = |V (H)|. The bound δlin-rem(H) ≤ r−2
+r−1 holds for every r-chromatic graph H; this follows
+from the Erdős-Simonovits supersaturation theorem, see by [12, Section 4.1] for the details.
+Suppose now that H contains no critical edge, and let us show that δlin-rem(H) ≥ r−2
+r−1. To this end, we
+construct, for every small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ r−2
+r−1n, such that
+G has at most O(ε2nh) copies of H, but Ω(εn2) edges must be deleted to turn G into an H-free graph. Let
+T (n, r − 1) be the Turán graph, i.e. the complete (r − 1)-partite graph with balanced parts V1, . . . , Vr−1.
+Add an εn-regular graph inside V1 and let the resulting graph be G. We first claim that G contains O(ε2nh)
+copies of H. As H contains no critical edge and χ(H) = r, every copy of H in G contains two edges e and
+e′ inside V1. If e and e′ are disjoint, then there are at most n2(εn)2 = ε2n4 choices for e and e′ and then at
+most nh−4 choices for the other h− 4 vertices of H. Therefore, there are at most ε2nh such H-copies. And if
+e and e′ intersect, then there are at most n(εn)2 = ε2n3 choices for e and e′ and then at most nh−3 choices
+for the remaining vertices, again giving at most ε2nh such H-copies. So G indeed has O(ε2nh) copies of H.
+On the other hand, we claim that one must delete Ω(εn2) edges to destroy all H-copies in G. Observe
+that G has at least 1
+2 |V1|·εn·|V2|·· · ··|Vr−1| = Ωr(εnr) copies of Kr, and every edge participates in at most
+nr−2 of these copies. Thus, deleting cεn2 edges can destroy at most cεnr copies of Kr. If c is a small enough
+constant (depending on r), then after deleting any cεn2 edges, there are still Ω(εnr) copies of Kr. Then,
+by Lemma 2.1, the remaining graph contains Kr[h], the h-blowup of Kr, and hence H. This completes the
+proof that δlin-rem(H) ≥ r−2
+r−1.
+We now prove that δlin-rem(H) ≥ 3r−8
+3r−5 for every r-chromatic graph H. It suffices to construct, for every
+small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ 3r−8
+3r−5n, such that G has at most
+O(ε2nh) copies of H but at least Ω(εn2) edges must be deleted to turn G into an H-free graph. The vertex
+set of G consists of r + 1 disjoint sets V0, V1, V2, . . . , Vr, where |Vi| =
+n
+3r−5 for i = 0, 1, 2, 3 and |Vi| =
+3n
+3r−5
+for i = 4, 5, . . ., r. Put complete bipartite graphs between V0 and V1, between V0 ∪ V1 and V4 ∪ · · · ∪ Vr, and
+between Vi to Vj for all 2 ≤ i < j ≤ r. Put εn-regular bipartite graphs between V1 and V2, and between V1
+and V3. The resulting graph is G (see Figure 2). It is easy check that δ(G) ≥ 3r−8
+3r−5n. Indeed, let 0 ≤ i ≤ r
+and v ∈ Vi. If 4 ≤ i ≤ r then v is connected to all vertices except for Vi; if i ∈ {2, 3} then v is connected to
+all vertices except V0 ∪ V1 ∪ Vi; and if i ∈ {0, 1} then v is connected to all vertices except V2 ∪ V3 ∪ Vi. In
+any case, the neighborhood of v misses at most
+3n
+3r−5 vertices.
+We claim that G has at most O(ε2nh) copies of H. Indeed, observe that if we delete all edges between V1
+and V2 then the remaining graph is (r − 1)-colorable with coloring V1 ∪ V2, V0 ∪ V3, V4, . . . , Vr. Hence, every
+copy of H must contain an edge e between V1 and V2. Similarly, every copy of H must contain an edge e′
+between V1 and V3. If e, e′ are disjoint then there are at most n2(εn)2 = ε2n4 ways to choose e, e′ and then
+at most nh−4 ways to choose the remaining vertices of H. And if e and e′ intersect then there are at most
+n(εn)2 = ε2n3 ways to choose e, e′ and at most nh−3 for the remaining h − 3 vertices of H. In both cases,
+the number of H-copies is at most ε2nh, as required.
+Now we show that one must delete Ω(εn2) edges to destroy all copies of H in G. Observe that G has
+|V1| · (εn)2 · |V4| · · · · · |Vr| = Ω(ε2nr) copies of Kr between the sets V1, . . . , Vr. We claim that every edge f
+4
+
+V1
+V2
+V3
+V0
+V1
+V2
+V3
+V4
+V0
+Figure 2: Proof of Lemma 3.2, r = 3 (left) and r = 4 (right). Heavy edges indicate complete bipartite graphs
+while dashed edges indicate εn-regular bipartite graphs.
+participates in at most εnr−2 of these r-cliques. Indeed, by the same argument as above, every copy of Kr
+containing f must contain an edge e from E(V1, V2) and an edge e′ from E(V1, V3). Suppose without loss of
+generality that e ̸= f (the case e′ ̸= f is symmetric). In the case f ∩ e = ∅, there are at most n · εn = εn2
+choices for e and at most nr−4 choices for the remaining vertices of Kr, giving at most εnr−2 copies of Kr
+containing f. And if f, e intersect, then there are at most εn choices for e and at most nr−3 for the remaining
+r − 3 vertices, giving again εnr−2.
+We see that deleting cεn2 edges of G can destroy at most cε2nr copies of Kr. Hence, if c is a small
+enough constant, then after deleting any cεn2 edges there are still Ω(ε2nr) copies of Kr left. By Lemma 2.1,
+the remaining graph contains a copy of Kr[h] and hence H. This completes the proof.
+4
+Polynomial removal thresholds: Proof of Theorem 1.4
+We say that an n-vertex graph G is ε-far from a graph property P (e.g. being H-free for a given graph H, or
+being homomorphic to a given graph F) if one must delete at least εn2 edges to make G satisfy P. Trivially,
+if G has εn2 edge-disjoint copies of H, then it is ε-far from being H-free. We need the following result from
+[21].
+Theorem 4.1. For every graph F on f vertices and for every ε > 0, there is q = qF (ε) = poly(f/ε), such
+that the following holds. If a graph G is ε-far from being homomorphic to F, then for a sample of q vertices
+x1, . . . , xq ∈ V (G), taken uniformly with repetitions, it holds that G[{x1, . . . , xq}] is not homomorphic to F
+with probability at least 2
+3.
+Theorem 4.1 is proved in Section 2 of [21].
+In fact, [21] proves a more general result on property
+testing of the so-called 0/1-partition properties. Such a property is given by an integer f and a function
+d : [f]2 → {0, 1, ⊥}, and a graph G satisfies the property if it has a partition V (G) = V1 ∪ · · · ∪ Vf such
+that for every 1 ≤ i, j ≤ f (possibly i = j), it holds that (Vi, Vj) is complete if d(i, j) = 1 and (Vi, Vj) is
+empty if d(i, j) = 0 (if d(i, j) =⊥ then there are no restrictions). One can express the property of having a
+homomorphism into F in this language, simply by setting d(i, j) = 0 for i = j and ij /∈ E(F). In [21], the
+class of these partition properties is denoted GPP0,1, and every such property is shown to be testable by
+sampling poly(f/ε) vertices. This implies Theorem 4.1.
+Proof of Theorem 1.4. Recall that IH is the set of minimal graphs H′ (with respect to inclusion) such that
+H is homomorphic to H′. For convenience, put δ := δhom(IH). Our goal is to show that δpoly-rem(H) ≤ δ+α
+for every α > 0. So fix α > 0 and let G be a graph with minimum degree δ(G) ≥ (δ + α)n and with
+εn2 edge-disjoint copies of H. By the definition of the homomorphism threshold, there is an IH-free graph
+F (depending only on IH and α) such that if a graph G0 is IH-free and has minimum degree at least
+(δ + α
+2 ) · |V (G0)|, then G0 is homomorphic to F. Observe that if a graph G0 is homomorphic to F then
+G0 is H-free, because F is free of any homomorphic image of H. It follows that G is ε-far from being
+homomorphic to F, because G is ε-far from being H-free. Now we apply Theorem 4.1. Let q = qF (ε) be
+given by Theorem 4.1. We assume that q ≫ log(1/α)
+α2
+and n ≫ q2 without loss of generality. Sample q vertices
+x1, . . . , xq ∈ V (G) with repetition and let X = {x1, . . . , xq}. By Theorem 4.1, G[X] is not homomorphic to
+F with probability at least 2/3. As n ≫ q2, the vertices x1, . . . , xq are pairwise-distinct with probability at
+least 0.99. Also, for every i ∈ [q], the number of indices j ∈ [q] \ {i} with xixj ∈ E(G) dominates a binomial
+5
+
+distribution B(q − 1, δ(G)
+n ). By the Chernoff bound (see e.g. [3, Appendix A]) and as δ(G) ≥ (δ + α)n,
+the number of such indices is at least (δ + α
+2 )q with probability 1 − e−Ω(qα2). Taking the union bound over
+i ∈ [q], we get that δ(G[X]) ≥ (δ + α
+2 )|X| with probability at least 1 − qe−Ω(qα2) ≥ 0.9, as q ≫ log(1/α)
+α2
+.
+Hence, with probability at least 1
+2 it holds that δ(G[X]) ≥ (δ + α
+2 )|X| and G[X] is not homomorphic to F. If
+this happens, then G[X] is not IH-free (by the choice of F), hence G[X] contains a copy of some H′ ∈ IH.
+By averaging, there is H′ ∈ IH such that G[X] contains a copy of H′ with probability at least
+1
+2|IH|. Put
+k = |V (H′)| and let M be the number of copies of H′ in G. The probability that G[X] contains a copy of H′
+is at most M( q
+n)k. Using the fact that q = polyH,α( 1
+ε), we conclude that M ≥
+1
+2|IH| · ( n
+q )k ≥ polyH,α(ε)nk.
+As H → H′, there exists H′′, a blow-up of H′, such that H′′ have the same number of vertices as H, and
+that H ⊂ H′′. By Lemma 2.1 for H′ with Vi = V (G) for all i, there exist polyH,α(ε)nv(H′′) copies of H′′ in
+G, and thus polyH,α(ε)nv(H) copies of H. This completes the proof.
+5
+Linear removal thresholds: Proof of Theorem 1.3
+Here we prove the upper bounds in Theorem 1.3; the lower bounds were proved in Section 3. The first case
+of Theorem 1.3 follows from Lemma 3.2, so it remains to prove the other two cases. We begin with some
+preparation. For disjoint sets A1, . . . , Am, we write �
+i∈[m] Ai × Ai+1 to denote all pairs of vertices which
+have one endpoint in Ai and one in Ai+1 for some 1 ≤ i ≤ m, with subscripts always taken modulo m. So a
+graph G has a homomorphism to the cycle Cm if and only if there is a partition V (G) = A1 ∪ · · · ∪ Am with
+E(G) ⊆ �
+i∈[m] Ai × Ai+1.
+Lemma 5.1. Suppose H is a graph such that χ(H) = 3, H contains a critical edge xy, and odd-girth(H) ≥
+2k + 1. Then,
+• There is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 = {x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪
+(�
+i∈[3] Ai × Ai+1);
+• if k ≥ 2, there is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that A1 = {x}, A2 = {y} and
+E(H) ⊆ �
+i∈[2k+1] Ai × Ai+1. In particular, H is homomorphic to C2k+1.
+Proof. Write H′ = H − xy, so H′ is bipartite. Let V (H) = V (H′) = L ·∪ R be a bipartition of H′. As
+χ(H) = 3, x and y must both lie in the same side of the bipartition. Without loss of generality, assume that
+x, y ∈ L. For the first item, set A1 = {x}, A2 = {y}, A3 = R and B = L\{x, y}. Then every edge of G goes
+between B and A3 or between two of the sets A1, A2, A3, as required.
+Suppose now that k ≥ 2, i.e. odd-girth(H) = 2k + 1 ≥ 5. For 1 ≤ i ≤ k, let Xi be the set of vertices at
+distance (i − 1) from x in H′, and let Yi be the set of vertices at distance (i − 1) from y in H′. Note that
+X1 = {x} and Y1 = {y}. Also, Xi, Yi lie in L if i is odd and in R if i is even. Write
+L′ := L\
+k�
+i=1
+(Xi ∪ Yi),
+R′ := R\
+k�
+i=1
+(Xi ∪ Yi),
+We first claim that {X1, . . . , Xk, Y1, . . . , Yk, L′, R′} forms a partition of V (H).
+The sets X1, . . . , Xk are
+clearly pairwise-disjoint, and so are Y1, . . . , Yk. Also, all of these sets are disjoint from L′, R′ by definition.
+So we only need to check Xi and Yj are disjoint for every pair 1 ≤ i, j ≤ k. Suppose for contradiction that
+there exists u ∈ Xi ∩ Yj for some 1 ≤ i, j ≤ k. Then i ≡ j (mod 2), because otherwise Xi, Yj are contained
+in different parts of the bipartition L, R. By the definition of Xi and Yj, H′ has a path x = x1, x2, . . . , xi = u
+and a path y = y1, y2, . . . , yj = u. Then, x = x1, x2, . . . , xi = u = yj, yj−1, . . . , y1, y, x forms a closed walk of
+length i+j −1, which is odd as i ≡ j (mod 2). Hence, odd-girth(H) ≤ 2k−1, contradicting our assumption.
+By definition, there are no edges between Xi and Xj for j − i ≥ 2, and similarly for Yi, Yj. Also, there
+are no edges between L′ ∪ R′ and �k−1
+i=1 (Xi ∪ Yi) because the vertices in L′ ∪ R′ are at distance more than k
+to x, y. Moreover, if k is even then there are no edges between Xk ∪ Yk and R′, and if k is odd then there
+are no edges between Xk ∪ Yk and L′. Next, we show that there are no edges between Xi and Yj for any
+1 ≤ i, j ≤ k except (i, j) = (1, 1). Indeed, if i = j then e(Xi, Yj) = 0 because Xi, Yj are on the same side
+6
+
+x
+y
+X2
+Y2
+L′
+R′
+x
+y
+X2
+Y2
+X3
+Y3
+R′
+L′
+Figure 3: Proof of Lemma 5.1, k = 2 (left) and k = 3 (right). Edges indicate bipartite graphs where edges
+can be present.
+of the bipartition L, R. So suppose that i ̸= j, say i < j, and assume by contradiction that there is an
+edge uv with u ∈ Xi, v ∈ Yj. Then v is at distance at most i + 1 ≤ k from x, implying that Yj intersects
+X1 ∪ · · · ∪ Xi+1, a contradiction.
+Finally, we define the partition A1, . . . , A2k+1 that satisfies the assertion of the second item. If k is even
+then take A1, . . . , A2k+1 to be X1, Y1, . . . , Yk−1, Yk∪R′, L′, Xk, . . . , X2, and if k is odd then take A1, . . . , A2k+1
+to be X1, Y1, . . . , Yk−1, Yk ∪ L′, R′, Xk, . . . , X2. See Figure 3 for an illustration. By the above, in both cases
+it holds that E(H) ⊆ �
+i∈[2k+1] Ai × Ai+1, as required.
+For vertex u ∈ V (G), denote by NG(u) the neighborhood of u and let degG(u) = |NG(u)|. For vertices
+u, v ∈ V (G), denote by NG(u, v) the common neighborhood of u, v and let degG(u, v) = |NG(u, v)|.
+Lemma 5.2. Let H be a graph on h vertices such that χ(H) = 3 and H contains a critical edge xy. Let G
+be a graph on n vertices with δ(G) ≥ αn. Let ab ∈ E(G) such that degG(a, b) ≥ αn. Then, there are at least
+poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G).
+Proof. By the first item of Lemma 5.1, there is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 =
+{x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪ �
+i∈[3] Ai × Ai+1. Let s = |A3| and t = |B|. Each u ∈ NG(a, b) has at
+least αn − 2 ≥ αn
+2 neighbors not equal to a, b. Hence, there are at least 1
+2 · |NG(a, b)| · αn
+2 ≥ α2n2
+4
+edges uv
+with u ∈ NG(a, b) and v /∈ {a, b}. Applying Lemma 2.1 with H = K2, V1 = NG(a, b) and V2 = V (G)\{a, b},
+we see that there are poly(α)ns+t pairs of disjoint sets (S, T ) such that |S| = s, |T | = t, S ⊆ NG(a, b),
+a, b /∈ T , and S, T form a complete bipartite graph in G. Given any such pair, it is safe to map x to a, y to
+b, A3 to S and B to T to obtain an H-copy. Hence, G contains at least poly(α)ns+t = poly(α)nh−2 copies
+of H mapping xy to ab.
+Lemma 5.3. Let H be a graph on h vertices such that χ(H) = 3, H contains a critical edge xy, and
+odd-girth(H) ≥ 5. Let G be a graph on n vertices, let ab ∈ E(G), and suppose that there exists A ⊂ NG(a)
+and B ⊂ NG(b) such that |A| , |B| ≥ αn and |NG(a′, b′)| ≥ αn for all distinct a′ ∈ A and b′ ∈ B. Then there
+are at least poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G).
+Proof. By Lemma 5.1 (using odd-girth(H) ≥ 5), there exists a partition V (H) = A1 ·∪ · · · ·∪ A5 such that
+A1 = {x}, A2 = {y}, and E(H) ⊆ �
+i∈[5] Ai × Ai+1. Put si = |Ai| for i ∈ [5].
+There are at least (|A||B| − |A|)/2 ≥ α2n2/3 pairs {a′, b′} of distinct vertices with a′ ∈ A, b′ ∈ B
+(the factor of 2 is due to the fact that each pair in A ∩ B is counted twice).
+Each such pair a′, b′ has
+at least αn − 2 ≥ αn/2 common neighbors c′ /∈ {a, b}, by assumption.
+Therefore, there are at least
+α2n2
+3
+· αn
+2
+= α3n3
+6
+triples (a′, b′, c′) such that a′ ∈ A, b′ ∈ B, and c′ ̸= a, b is a common neighbor of a′, b′.
+By Lemma 2.1 with H = K2,1 and V1 = A, V2 = B, V3 = V (G)\{a, b}, there are at least poly(α)ns3+s4+s5
+corresponding copies of K2,1[s3, s5, s4], i.e., triples of disjoint sets (R, S, T ) such that R ⊆ A, S ⊆ B, a, b /∈ T ,
+|R| = s5, |S| = s3, |T | = s4, and (R, T ) and (S, T ) form complete bipartite graphs in G. Given any such
+7
+
+triple, we can safely map A1 = {x} to a, A2 = {y} to b, A5 to R, A3 to S, and A4 to T to obtain a copy of
+H. Thus, there are at least poly(α)ns3+s4+s5 = poly(α)nh−2 copies of H mapping xy to ab.
+In the following theorem we prove the upper bound in the second case of Theorem 1.3.
+Theorem 5.4. Let H be a graph such that χ(H) = 3, H has a critical edge xy, and H contains a triangle.
+Then, δlin-rem(H) ≤ 1
+3.
+Proof. Write h = v(H). Fix an arbitrary α > 0, and let G be an n-vertex graph with minimum degree
+δ(G) ≥ ( 1
+3 + α)n and with a collection C = {H1, . . . , Hm} of m := εn2 edge-disjoint copies of H.
+For
+each i = 1, . . . , m, there exist u, v, w ∈ V (Hi) forming a triangle (because H contains a triangle).
+As
+degG(u) + degG(v) + degG(w) ≥ 3δ(G) ≥ (1 + 3α)n, two of u, v, w have at least αn common neighbors. We
+denote these two vertices by ai and bi. By Lemma 5.2, G has at least poly(α)nh−2 copies of H which map
+xy to aibi. The edges a1b1, . . . , ambm are distinct because H1, . . . , Hm are edge-disjoint. Hence, summing
+over all i = 1, . . . , m, we see that G contains at least εn2 · poly(α)nh−2 = poly(α)εnh copies of H. This
+proves that δlin-rem(H) ≤ 1
+3 + α, and taking α → 0 gives δlin-rem(H) ≤ 1
+3.
+In what follows, we need the following very well-known observation, originating in the work of Andrásfai,
+Erdős and Sós, see [4, Remark 1.6].
+Lemma 5.5. If δ(G) >
+2
+2k+1n and odd-girth(G) ≥ 2k + 1 for k ≥ 2, then G is bipartite.
+Proof. Suppose by contradiction that G is not bipartite and take a shortest odd cycle C in G, so |C| ≥ 2k+1.
+As �
+x∈C deg(x) ≥ (2k+1)δ(G) > 2n, there exists a vertex v /∈ C with at least 3 neighbors on C. Then there
+are two neighbors x, y ∈ C of v such that the distance of x, y along C is not equal to 2. Then by taking the
+odd path between x, y along C and adding the edges vx, vy, we get a shorter odd cycle, a contradiction.
+We will also use the following result of Letzter and Snyder, see [17, Corollary 32].
+Theorem 5.6 ([17]). Let G be a {C3, C5}-free graph on n vertices with δ(G) > n
+4 . Then G is homomorphic
+to C7.
+We can now finally prove the upper bound in the last case of Theorem 1.3.
+Theorem 5.7. Let H be a graph such that χ(H) = 3, H contains critical edge xy, and odd-girth(H) ≥ 5.
+Then δlin-rem(H) ≤ 1
+4.
+Proof. Denote h = |V (H)|. Write odd-girth(G) = 2k + 1 ≥ 5. By the second item of Lemma 5.1, there
+is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that |A1| = |A2| = 1, and E(H) ⊆ �
+i∈[2k+1] Ai × Ai+1.
+Denote si = |Ai| for each i ∈ [2k + 1], so H is a subgraph of the blow-up C2k+1[s1, . . . , s2k+1] of C2k+1. Let
+c1 = c1(C2k+1, s1, . . . , s2k+1) > 0 and c2 = c2(k) > 0 be the constants given by Lemma 2.1 and Lemma 2.3,
+respectively. According to Theorem 1.2, δpoly-rem(C2k+1) =
+1
+2k+1 < 1
+4, and hence there exists a constant
+c3 = c3(k) > 0 such that if G is a graph on n vertices with δ(G) ≥
+n
+4 and at least εn2 edge-disjoint
+C2k+1-copies, then G contains at least c3ε
+1
+c3 n2k+1 copies of C2k+1. Set c := c1 · min(c2, c3).
+Let α > 0 and ε be small enough; it suffices to assume that ε <
+�
+α2
+200k(k+2)
+�1/c
+. Let G be a graph on
+n vertices with δ(G) ≥ ( 1
+4 + α)n which contains at least εn2 edge-disjoint copies of H. Our goal is to show
+that G contains ΩH,α(εnh) copies of H. Suppose first that G contains at least εcn2 edge-disjoint copies of
+C2ℓ+1 for some 1 ≤ ℓ ≤ k. If ℓ < k, then G contains Ωk(εc/c2n2k+1) = Ωk(εc1n2k+1) copies of C2k+1 by
+Lemma 2.3 and the choice of c2. And if ℓ = k, then G contains Ωk(εc/c3n2k+1) = Ωk(εc1n2k+1) copies of
+C2k+1 by Theorem 1.2 and the choice of c3. In either case, G contains Ωk(εc1n2k+1) copies of C2k+1. But
+then, by Lemma 2.1 (with V1 = · · · = V2k+1 = V (G)), G contains at least ΩH(εc1/c1nh) = ΩH(εnh) copies
+of C2k+1[s1, . . . , s2k+1], and hence ΩH,α(εnh) copies of H. This concludes the proof of this case.
+From now on, assume that G contains at most εcn2 edge-disjoint C2ℓ+1-copies for every ℓ ∈ [k]. Let Cℓ
+be a maximal collection of edge-disjoint C2ℓ+1-copies in G, so |Cℓ| ≤ εcn2. Let Ec be the set of edges which
+8
+
+are contained in one of the cycles in C1 ∪ · · · ∪ Ck. Let S be the set of vertices which are incident with at
+least αn
+10 edges from Ec. Then
+|Ec| ≤
+k
+�
+ℓ=1
+(2ℓ + 1)εcn2 = k(k + 2)εcn2 and |S| ≤ 2 |Ec|
+αn/10 ≤ 20k(k + 2)εc
+α
+n < αn
+10 ,
+(1)
+where the last inequality holds by our assumed bound on ε. Let G′ be the subgraph of G obtained by deleting
+the edges in Ec and the vertices in S. Note that G′ ⊆ G − Ec is {C3, C5, . . . , C2k+1}-free because for every
+1 ≤ ℓ ≤ k, we removed all edges from a maximal collection of edge-disjoint C2ℓ+1-copies.
+Claim 5.8. |V (G′)| > (1 − α
+10)n and δ(G′) > ( 1
+4 + 4α
+5 )n.
+Proof. The first inequality follows from (1) as |V (G′)| = n − |S|. Each v ∈ V (G)\S has at most αn
+10 incident
+edges from Ec, and at most |S| < αn
+10 neighbors in S, thereby degG′(v) > degG(v) − αn
+5 ≥ ( 1
+4 + 4α
+5 )n. Hence,
+δ(G′) > ( 1
+4 + 4α
+5 )n.
+Claim 5.9. G′ is homomorphic to C7. Moreover, G′ is bipartite unless k = 2.
+Proof. Recall that G′ is {C3, C5, . . . , C2k+1}-free. As k ≥ 2, G′ is {C3, C5}-free. Also, δ(G′) > n
+4 ≥ |V (G′)|
+4
+by Claim 5.8.
+So G′ is homomorphic to C7 by Theorem 5.6.
+If k ≥ 3, i.e.
+odd-girth(H) ≥ 7, then
+odd-girth(G′) ≥ 2k + 3 ≥ 9. As δ(G′) > n
+4 , G′ is bipartite by Lemma 5.5.
+The rest of the proof is divided into two cases based whether or not G′ is bipartite. These cases are handled
+by Propositions 5.10 and 5.11, respectively.
+Proposition 5.10. Suppose that G′ is bipartite. Then G has ΩH,α(εnh) copies of H.
+Proof. Let (L′, R′) be a bipartition of G′, so V (G) = L′ ·∪ R′ ·∪ S. Let L1 ⊆ S (resp. R1 ⊆ S) be the set of
+vertices of S having at most αn
+5 neighbors in L′ (resp. R′). Let G′′ be the bipartite subgraph of G induced
+by the bipartition (L′′, R′′) := (L′ ·∪ L1, R′ ·∪ R1). Let S′′ = V (G)\(L′′ ·∪ R′′), so V (G) = L′′ ·∪ R′′ ·∪ S′′.
+We claim that δ(G′′) ≥ ( 1
+4 + α
+2 )n. First, as G′ is a subgraph of G′′, we have degG′′(v) > ( 1
+4 + 4α
+5 )n for
+each v ∈ V (G′) ⊆ V (G′′) by Claim 5.8. Now we consider vertices in V (G′′) \ V (G′) = L1 ∪ R1. Each v ∈ L1
+has at most |S| ≤ αn
+10 neighbors in S and at most αn
+5
+neighbors in L′, by the definition of L1. Hence, v
+has at least degG(v) − 3α
+10 n ≥ ( 1
+4 + α
+2 )n neighbors in R′ ⊆ V (G′′). By the symmetric argument for vertices
+v ∈ R1, we get that δ(G′′) ≥ ( 1
+4 + α
+2 )n, as required.
+For an edge uv ∈ E(G)\E(G′′), we say uv is of type I if u, v ∈ L′′ or u, v ∈ R′′, and we say that uv is
+of type II if u ∈ S′′ or v ∈ S′′. Every edge in E(G)\E(G′′) is of type I or II. Since χ(H) = 3 and G′′ is
+bipartite, each copy of H in G must contain an edge of type I or an edge of type II (or both). As G has
+εn2 edge-disjoint H-copies, G contains at least εn2
+2
+edges of type I or at least εn2
+2
+edges of type II. We now
+consider these two cases separately. See Fig. 4 for an illustration. Recall that xy ∈ E(H) denotes a critical
+edge of H.
+Case 1:
+G contains εn2
+2
+edges of type I.
+Fix any edge ab ∈ E(G) of type I. Without loss of generality,
+assume a, b ∈ L′′ (the case a, b ∈ R′′ is symmetric). We claim that G has poly(α)nh−2 copies of H mapping
+xy ∈ E(H) to ab ∈ E(G). If degG(a, b) ≥ αn
+2 then this holds by Lemma 5.2. Otherwise, degG(a, b) < αn
+2 ,
+and thus
+|R′′| ≥ |NG′′(a) ∪ NG′′(b)| ≥ degG′′(a) + degG′′(b) − degG(a, b) > 2δ(G′′) − αn
+2 > n
+2 ,
+using that δ(G′′) ≥ ( 1
+4 + α
+2 )n. Thus, |L′′| < n
+2 . This implies that for all a′ ∈ NG′′(a), b′ ∈ NG′′(b),
+degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn.
+Now, by Lemma 5.3 (with A = NG′′(a) and B = NG′′(b)), there are poly(α)nh−2 copies of H mapping xy
+to ab, as claimed. Summing over all edges ab of type I, we get εn2
+2 · poly(α)nh−2 = poly(α)εnh copies of H.
+This completes the proof in Case 1.
+9
+
+L′′
+R′′
+a
+b
+L′′
+R′′
+a
+b
+a′
+b′
+L′′
+R′′
+S′′
+a
+b
+a′
+b′
+Figure 4: Proof of Proposition 5.10: Case 1 with degG(a, b) ≥
+αn
+2
+(left), Case 1 with degG(a, b) <
+αn
+2
+(middle) and Case 2 (right). The red part is the common neighborhood of a and b (or a′ and b′).
+Case 2:
+G contains εn2
+2
+edges of type II.
+Note that the number of edges of type II is trivially at most
+|S′′| n.
+Thus, |S′′| ≥
+εn
+2 .
+Fix some a ∈ S′′.
+By the definition of L′′, R′′ and S′′, v has at least
+αn
+5
+neighbors in L′ ⊆ L′′ and at least αn
+5 neighbors in R′ ⊆ R′′. Without loss of generality, assume |L′′| ≤ |R′′|,
+thereby |L′′| ≤
+n
+2 .
+Now fix any b ∈ L′′ adjacent to a; there are at least
+αn
+5
+choices for b.
+We have
+|NG(a) ∩ R′′| ≥ αn
+5 and |NG′′(b)| ≥ δ(G′′) > n
+4 , and for all a′ ∈ NG(a) ∩ R′′, b′ ∈ NG′′(b) ⊆ R′′ it holds that
+degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn. Therefore, by Lemma 5.3, G has poly(α)nh−2 copies of H mapping xy
+to ab. Enumerating over all a ∈ S′′ and b ∈ NG(a) ∩ L′′, we again get ΩH,α(εnh) copies of H in G. This
+completes the proof of Proposition 5.10.
+Proposition 5.11. Suppose G′ is non-bipartite but homomorphic to C7. Then G has ΩH,α(εnh) copies of H.
+Proof. By Claim 5.9 we must have k = 2 , so odd-girth(H) = 5. The proof is similar to that of Proposi-
+tion 5.10, but instead of a bipartition of G′, we use a partition corresponding to a homomorphism into C7.
+Let V (G)\S = V (G′) = V ′
+1 ·∪ V ′
+2 ·∪ · · · ·∪ V ′
+7 be a partition of V (G′) such that E(G′) ⊆ �
+i∈[7] V ′
+i × V ′
+i+1.
+Here and later, all subscripts are modulo 7. We have V ′
+i ̸= ∅ for all i ∈ [7], because otherwise G′ would be
+bipartite. For i ∈ [7], let Si be the set of vertices in S having at most 2αn
+5
+neighbors in V (G′)\ (V ′
+i−1 ∪V ′
+i+1).
+In case v lies in multiple Si’s, we put v arbitrarily in one of them. Set V ′′
+i
+:= V ′
+i ∪ Si. Let G′′ be the
+7-partite subgraph of G with parts V ′′
+1 , . . . , V ′′
+7 and with all edges of G between V ′′
+i
+and V ′′
+i+1, i = 1, . . . , 7.
+By definition, G′ is a subgraph of G′′, and G′′ is homomorphic to C7 via the homomorphism V ′′
+i �→ i. Put
+S′′ := V (G)\V (G′′) = S \ �7
+i=1 Si. We now collect the following useful properties.
+Claim 5.12. The following holds:
+(i) δ(G′′) ≥ ( 1
+4 + α
+2 )n.
+(ii) For every i ∈ [7] and for every u, v ∈ V ′′
+i
+or u ∈ V ′′
+i , v ∈ V ′′
+i+2, it holds that degG′′(u, v) ≥ αn.
+(iii) For every i ∈ [7], every v ∈ V ′′
+i
+has at least αn neighbors in V ′′
+i−1 and at least αn neighbors in V ′′
+i+1.
+(iv) For every a ∈ S′′, there are i, j with j − i ≡ 1, 3 (mod 7) and |NG(a) ∩ V ′′
+i | ,
+��NG(a) ∩ V ′′
+j
+�� > 2αn
+25 .
+Proof. fds
+(i) Let i ∈ [7] and v ∈ V ′′
+i . If v ∈ V (G′), then degG′′(v) ≥ degG′(v) ≥ δ(G′) > ( 1
+4 + α
+2 )n, using Claim 5.8.
+Otherwise, v ∈ Si. By definition, v has at most 2αn
+5
+neighbours in V (G′)\(V ′
+i−1 ∪V ′
+i+1). Also, v has at
+most |S| ≤ αn
+10 neighbours in S. It follows that v has at least degG(v)− 2αn
+5 − αn
+10 ≥ ( 1
+4 + α
+2 )n neighbors
+in V ′′
+i−1 ∪ V ′′
+i+1. Hence, degG′′(v) > ( 1
+4 + α
+2 )n.
+(ii) First, observe that
+|V ′′
+i | +
+��V ′′
+i+2
+�� ≥
+�1
+4 + α
+2
+�
+n
+(2)
+10
+
+for all i ∈ [7]. Indeed, V ′′
+i+1 is non-empty, and fixing any v ∈ V ′′
+i+1, we have |V ′′
+i | +
+��V ′′
+i+2
+�� ≥ degG′′(v) ≥
+δ(G′′) ≥ ( 1
+4 + α
+2 )n. By applying (2) to the pairs (i + 2, i + 4) and (i − 2, i), we get
+��V ′′
+i−1
+�� +
+��V ′′
+i+1
+�� +
+��V ′′
+i+3
+�� ≤ n − (
+��V ′′
+i+2
+�� +
+��V ′′
+i+4
+��) − (
+��V ′′
+i−2
+�� + |V ′′
+i |) ≤ n − 2
+�1
+4 + α
+2
+�
+n < n
+2 .
+(3)
+Now let i ∈ [7]. For u, v ∈ V ′′
+i
+we have NG′′(u) ∪ NG′′(v) ⊆ V ′′
+i−1 ∪ V ′′
+i+1, and for u ∈ V ′′
+i , v ∈ V ′′
+i+2
+we have NG′′(u) ∪ NG′′(v) ⊆ V ′′
+i−1 ∪ V ′′
+i+1 ∪ V ′′
+i+3. In both cases, |NG′′(u) ∪ NG′′(v)| < n
+2 by (3). As
+degG′′(u) + degG′′(v) ≥ 2δ(G′′) ≥ ( 1
+2 + α)n, we have degG′′(u, v) > αn, as required.
+(iii) We first argue that |V ′′
+i | ≤ ( 1
+4 − 3α
+2 )n for each i ∈ [7]. Indeed, by applying (2) to the pairs (i − 1, i + 1),
+(i + 2, i + 4), (i + 3, i + 5), we get
+|V ′′
+i | ≤ n − (
+��V ′′
+i−1
+�� +
+��V ′′
+i+1
+��) − (
+��V ′′
+i+2
+�� +
+��V ′′
+i+4
+��) − (
+��V ′′
+i+3
+�� +
+��V ′′
+i+5
+��) ≤ n − 3
+�1
+4 + α
+2
+�
+n =
+�1
+4 − 3α
+2
+�
+n.
+Now, for every v ∈ V ′′
+i , we have NG′′(v) ⊆ V ′′
+i−1 ∪ V ′′
+i+1 and
+��V ′′
+i−1
+�� ,
+��V ′′
+i+1
+�� < ( 1
+4 − 3α
+2 )n. Hence, v has
+at least degG′′(v) − ( 1
+4 − 3α
+2 )n ≥ αn neighbors in each of V ′′
+i−1, V ′′
+i+1.
+(iv) Let I be the set of i with |NG(a) ∩ V ′′
+i | ≥ 2αn
+25 . If I is empty, then a has less than 5 · 2αn
+25
+= 2αn
+5
+neighbors in every V (G′)\(V ′
+i−1 ∪V ′
+i+1) and therefore can not be in S′′. Suppose for contradiction that
+there exist no i, j ∈ I with j − i ≡ 1, 3 (mod 7). We claim that there is j ∈ [7] such that I ⊆ {j, j + 2}.
+Fix an arbitrary i ∈ I. Then, i ± 1, i ± 3 /∈ I by assumption. Also, at most one of i + 2, i − 2 is
+in I, because (i − 2) − (i + 2) ≡ 3 (mod 7). So I ⊆ {i, i + 2} or I ⊆ {i − 2, i}, proving our claim
+that I ⊆ {j, j + 2} for some j. By the definition of I, a has at most 5 · 2αn
+25
+=
+2αn
+5
+neighbors in
+V (G′)\(V ′
+j ∪ V ′
+j+2). Hence, a ∈ Sj+1. This contradicts the fact that a ∈ S′′, as S′′ ∩ Si+1 = ∅.
+We continue with the proof of Proposition 5.11. Recall that the edges in E(G) \ E(G′′) are precisely
+the edges of G not belonging to �
+i∈[7] V ′′
+i × V ′′
+i+1. For an edge ab ∈ E(G)\E(G′′), we say ab is of type I if
+a, b ∈ V (G′′), and of type II if a ∈ S′′ or b ∈ S′′. Clearly, every edge in E(G)\E(G′′) is either of type I
+or of type II. Since odd-girth(H) = 5 and C5 is not homomorphic to C7, every H-copy in G must contain
+some edge of type I or of type II (or both). As G has εn2 edge-disjoint H-copies, G must have at least εn2
+2
+edges of type I or at least εn2
+2
+edges of type II. We consider these two cases separately. See Fig. 5 for an
+illustration. Recall that xy ∈ E(H) denotes a critical edge of H.
+Case 1:
+G contains εn2
+2
+edges of type I. Fix any edge ab of type I, where a ∈ V ′′
+i and b ∈ V ′′
+j for i, j ∈ [7].
+We now show that G has poly(α)nh−2 copies of H mapping xy ∈ E(H) to ab. As ab /∈ E(G′′), we have
+i−j ≡ 0, ±2, ±3 (mod 7). When j−i ≡ 0, ±2 (mod 7), we have degG(a, b) ≥ degG′′(a, b) > αn by Claim 5.12
+(ii). Then, by Lemma 5.2, G has poly(α)nh−2 copies of H mapping xy to ab, as required. Now suppose that
+j−i ≡ ±3 (mod 7), say j ≡ i+3 (mod 7). Denote A := NG(a)∩V ′′
+i−1 and B := NG(b)∩V ′′
+j+1 = NG(b)∩V ′′
+i−3.
+We have that |A| , |B| ≥ αn by Claim 5.12 (iii), and |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B by Claim 5.12
+(ii). Now, by Lemma 5.3, G has poly(α)nh−2 copies of H mapping xy to ab, proving our claim. Summing
+over all edges ab of type I, we get εn2
+2 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G, finishing this case.
+Case 2:
+G contains εn2
+2
+edges of type II. Notice that the number edges incident to S′′ is at most |S′′| n,
+meaning that |S′′| ≥ εn
+2 . Fix any a ∈ S′′. By Claim 5.12 (iv), there exist i, j ∈ [7] with j − i ≡ 1, 3 (mod 7)
+and |NG(a) ∩ V ′′
+i | ,
+��NG(a) ∩ V ′′
+j
+�� > 2αn
+25 . Fix any b ∈ NG(a) ∩ V ′′
+i (there are at least 2αn
+25 choices for b). Take
+A = NG(a)∩V ′′
+j and B = NG(b)∩V ′′
+i+1. We have that |A| ≥ 2αn
+25 , and |B| ≥ αn by Claim 5.12 (iii). Further,
+as j − (i + 1) ≡ 0, 2 (mod 7), Claim 5.12 (ii) implies that |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B. Now,
+by Lemma 5.3, G has poly(α)nh−2 copies of H mapping xy to ab. Summing over all choices of a ∈ S′′ and
+b ∈ V ′′
+i , we acquire |S′′| · 2αn
+25 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G. This completes the proof of Case
+2, and hence the proposition.
+Propositions 5.10 and 5.11 imply the theorem.
+11
+
+V ′′
+1
+V ′′
+2
+V ′′
+3
+V ′′
+4
+V ′′
+5
+V ′′
+6
+V ′′
+7
+a
+b
+V ′′
+1
+V ′′
+2
+V ′′
+3
+V ′′
+4
+V ′′
+5
+V ′′
+6
+V ′′
+7
+a
+b
+a′
+b′
+V ′′
+1
+V ′′
+2
+V ′′
+3
+V ′′
+4
+V ′′
+5
+V ′′
+6
+V ′′
+7
+S′′ a
+b
+a′
+b′
+Figure 5: Proof of Proposition 5.11: Case 1 for j = i + 2 (left), Case 1 for j = i + 3 (middle) and Case 2 for
+j = i + 3 (right). The red part is the common neighborhood of a and b (or a′ and b′).
+6
+Concluding remarks and open questions
+It would be interesting to determine the possible values of δpoly-rem(H) for 3-chromatic graphs H. So far we
+know that
+1
+2k+1 is a value for each k ≥ 1. Is there a graph H with 1
+5 < δpoly-rem(H) < 1
+3? Also, is it true
+that δpoly-rem(H) > 1
+5 if H is not homomorphic to C5?
+Another question is whether the inequality in Theorem 1.4 is always tight, i.e. is it always true that
+δpoly-rem(H) = δhom(IH)?
+Finally, we wonder whether the parameters δpoly-rem(H) and δlin-rem(H) are monotone, in the sense that
+they do not increase when passing to a subgraph of H. We are not aware of a way of proving this without
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+13
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+L-HYDRA: MULTI-HEAD PHYSICS-INFORMED NEURAL
+NETWORKS
+ZONGREN ZOU∗ AND GEORGE EM KARNIADAKIS†
+Abstract. We introduce multi-head neural networks (MH-NNs) to physics-informed machine
+learning, which is a type of neural networks (NNs) with all nonlinear hidden layers as the body and
+multiple linear output layers as multi-head. Hence, we construct multi-head physics-informed neural
+networks (MH-PINNs) as a potent tool for multi-task learning (MTL), generative modeling, and
+few-shot learning for diverse problems in scientific machine learning (SciML). MH-PINNs connect
+multiple functions/tasks via a shared body as the basis functions as well as a shared distribution
+for the head. The former is accomplished by solving multiple tasks with MH-PINNs with each head
+independently corresponding to each task, while the latter by employing normalizing flows (NFs) for
+density estimate and generative modeling. To this end, our method is a two-stage method, and both
+stages can be tackled with standard deep learning tools of NNs, enabling easy implementation in
+practice. MH-PINNs can be used for various purposes, such as approximating stochastic processes,
+solving multiple tasks synergistically, providing informative prior knowledge for downstream few-shot
+learning tasks such as meta-learning and transfer learning, learning representative basis functions,
+and uncertainty quantification. We demonstrate the effectiveness of MH-PINNs in five benchmarks,
+investigating also the possibility of synergistic learning in regression analysis. We name the open-
+source code “Lernaean Hydra” (L-HYDRA), since this mythical creature possessed many heads for
+performing important multiple tasks, as in the proposed method.
+Key words.
+PINNs, meta-learning, multi-tasking, transfer learning, generative models, nor-
+malizing flows, stochastic problems
+MSC codes. 34F05, 62M45, 65L99, 65M99, 65N99
+1. Introduction. Learning across tasks has drawn great attention recently in
+deep learning and is an emerging theme in scientific machine learning (SciML), due
+to the fact that several classes of scientific problems are similar and/or related in-
+trinsically by their common physics.
+Intuitively, if tasks are similar, e.g., in the
+context of approximating stochastic processes [44], learning solution operators of ordi-
+nary/partial differential equations (ODEs/PDEs) [28], and solving parametric PDEs
+[42, 19, 4], it may be beneficial to relate them in the modeling, algorithm design,
+and/or solving procedure. In this regard, machine learning solvers, developed rapidly
+in the past few years, are considerably more flexible and of higher potential compared
+to traditional numerical solvers. Significant progress has been witnessed in the general
+area, including meta-learning for solving ODEs/PDEs [30, 27, 34, 6], transfer learning
+for physics-informed neural networks (PINNs) [3, 7], transfer learning for domain shift
+in solving PDEs [14], multi-task learning for PINNs [40], and generative methods for
+solving stochastic differential equations (SDEs) [44, 46, 15]. More recently, operator
+learning [28, 24] in which direct operator mapping is learned and subsequently used
+for other tasks in one-shot format has attracted a lot of attention.
+Multi-head neural networks (MH-NNs) fit perfectly different scenarios of learning
+across tasks. They were originally proposed as members of hard-parameter sharing
+neural networks (NNs) for deep multi-task learning (MTL) [5], in which multiple
+tasks, denoted as Tk, k = 1, ..., M, where M is the number of total tasks, are solved
+simultaneously. The general goals of using MH-NNs in MTL are diverse: achieving
+∗Division of Applied Mathematics,
+Brown University,
+Providence,
+RI 02912,
+USA (zon-
+gren zou@brown.edu).
+†Corresponding author.
+Division of Applied Mathematics, Brown University, Providence, RI
+02912, USA (george karniadakis@brown.edu).
+1
+arXiv:2301.02152v1  [cs.LG]  5 Jan 2023
+
+2
+Z. ZOU AND G. E. KARNIADAKIS
+better performance for all tasks, learning good and useful representations for down-
+stream tasks, and/or boosting the learning of main tasks with the help of auxiliary
+tasks. Moreover, although originally designed for solving multiple tasks, MH-NNs in
+recent years have also been extensively used for meta-learning. For example, in [41],
+the connection between MTL and meta-learning was analyzed, and meta-learning al-
+gorithms for MH-NN were discussed; in [25], it was shown that MH-NNs, trained in
+MTL fashion also perform task-specific adaptation in meta-learning; [37] argued that
+the effectiveness of model-agnostic meta-learning [10], a well-known meta-learning al-
+gorithm, may be due to successfully learned good representations rather than learned
+adaptation, and MH-NNs were used to study the detailed contributions of NNs in
+fast task adaptations. Overall, it is commonly acknowledged in the literature that
+when used to solve previous tasks, MH-NNs are capable of distilling useful shared
+information and storing it in their bodies and heads.
+In this paper, we develop MH-NNs for physics-informed machine learning [17],
+propose multi-head physics-informed neural networks (MH-PINNs), and further in-
+vestigate their applicability and capabilities to MTL, generative modeling, and meta-
+learning. A MH-PINN, as shown in Fig. 1, is built upon a conventional MH-NN and
+consists of two main parts, the body and multiple heads, and each head connects to
+a specific ODE/PDE task. Many architecture splitting strategies for MH-NNs are
+adopted in different applications scenarios; e.g., for some computer vision problems,
+a NN is split such that the body consists of convolutional layers and is followed by
+fully-connected layers as heads. In this paper, however, we choose the simplest one,
+i.e., the body consists of all nonlinear layers and the head is the last linear layer,
+for the following two reasons: (1) the dimensionality of the head is reduced, which
+enables fast density estimation (see next section); and (2) the body spontaneously
+provides a set of basis functions.
+Fig. 1. Schematic view of the structure of multi-head physics-informed neural networks (MH-
+PINNs) with M different heads, which are built upon conventional multi-head neural networks.
+The shared layers are often referred to as body and the task-specific layer as head.
+Generally,
+uk, k = 1, ..., M represent M solutions to M different ODEs/PDEs, formulated in Eq. (2.1), which
+may differ in source terms fk, boundary/initial condition terms bk, or differential operator Fk.
+The novelty and major contributions of this work are as follows:
+1. We propose a new physics-informed generative method using MH-PINNs for
+learning stochastic processes from data and physics.
+2. We propose a new method for physics-informed few-shot regression problems
+with uncertainty quantification using MH-PINNs.
+3. We study and demonstrate the effectiveness of MTL and synergistic learning
+with MH-NNs in regression problems.
+The paper is organized as follows. In Sec. 2, we present the problem formulation,
+
+Fiui(α)/ = fi(α), Biui(α)/ = bi(α)
+head
+task,
+1
+F2[u2(α)] = f2(α), B2[u2(α)] = b2(α)
+head,
+u2
+Body
+-
+FM[uM(α)] = fM(α), BM[uM(c)] = bM(c)
+head
+uM
+task,
+ML-HYDRA
+3
+details of MH-PINNs, and the general methodology, including how to use MH-PINNs
+for MTL, generative modeling, downstream few-shot physics-informed learning with
+uncertainty quantification (UQ). In Sec. 3, we discuss existing research closely related
+to our work and compare them conceptually. In Sec. 4, we test MH-PINNs with five
+benchmarks, each of which corresponds to one or more learning purposes, e.g., MTL
+and generative modeling.
+In Sec. 5, we investigate MTL and synergistic learning
+with the function approximation example. We conclude and summarize in Sec. 6.
+The details of our experiments, such as NN architectures and training strategies, can
+be found in Appendix A and B, as well as in the L-HYDRA open-source codes on
+GitHub, which will be released once the paper is accepted.
+2. Methodology. We assume that we have a family of tasks, {Tk}M
+k=1, each of
+which is associated with data Dk, k = 1, ..., M. The primary focus of this paper is on
+scientific computing and ODEs/PDEs, and therefore we further assume {Tk}M
+k=1 are
+physics-informed regression problems [17].
+Consider a PDE of the following form:
+Fk[uk(x)] = fk(x), x ∈ Ωk,
+(2.1a)
+Bk[uk(x)] = bk(x), x ∈ ∂Ωk,
+(2.1b)
+where k denotes the index of the task and k = 1, ..., M, x is the general spatial-
+temporal coordinate of Dx dimensions, Ωk are bounded domains, fk and uk are the
+Du-dimensional source terms and solutions to the PDE, respectively, Fk are general
+differential operators, Bk are general boundary/initial condition operators, and bk are
+boundary/initial condition terms. For simplicity, throughout this paper, the domain
+and the boundary/initial operator, denoted as Ω and B, are assumed to be the same
+for all tasks, and the solutions uk to be task-specific. The task Tk is described as
+approximating uk, and/or fk, and/or Fk, and/or bk, from data Dk and Eq. (2.1).
+Traditional numerical solvers often tackle {Tk}M
+k=1 independently, without lever-
+aging or transferring knowledge across tasks. The PINN method [38] was designed to
+solve ODEs/PDEs independently using NNs, which, however, yields M uncorrelated
+results. In this paper instead we treat {Tk}M
+k=1 as a whole and connect them with MH-
+PINNs, the architecture of which, shown in Fig. 1, enforces basis-functions-sharing
+predictions on the solutions uk. In addition to the informative representation/body,
+we further relate {Tk}M
+k=1 by assuming that their corresponding heads in MH-PINNs,
+denoted as {Hk}M
+k=1, are samples of a random variable with unknown probability
+density function (PDF), denoted as H and p(H), respectively. The shared body and
+a generative model of H immediately form a generative model of the solution u, and
+generators of the source term f and the boundary/initial term b as well by substitut-
+ing u into Eq. (2.1) and automatic differentiation [1], from which a generative method
+for approximating stochastic processes is seamlessly developed.
+Generators of u, f and b, as discussed in [30], are able to provide an informative
+prior distribution in physics-informed Bayesian inference [43, 26] as well as in UQ
+for SciML [47, 36], where the informative prior compensates for the insufficiency of
+observational data to address the physics-informed learning problems with even a few
+noisy measurements. In this paper, we generalize such problem to deterministic cases
+as well, where the data is noiseless and methods and results are deterministic, and
+refer to it as few-shot physics-informed learning. The general idea is to apply prior
+knowledge learned from connecting {Tk}M
+k=1 with MH-PINNs to new tasks, denoted
+as ˜T , associated with insufficient data ˜D, for accurate and trustworthy predictions.
+
+4
+Z. ZOU AND G. E. KARNIADAKIS
+The schematic view of the learning framework is illustrated in Fig. 2, and the details
+are explained next.
+Fig. 2. Schematic view of the learning framework and the proposed method. Three general
+types of learning are addressed: physics-informed learning, generative modeling, and few-shot learn-
+ing. The physics-informed learning is performed with MH-PINNs; the generative modeling is done
+afterwards by density estimate over the head via normalizing flows (NFs); in the end the few-shot
+physics-informed learning is accomplished with prior knowledge obtained from previous two via either
+fine-tuning with the learned regularization or Bayesian inference with the learned prior distribution.
+The body represents the set of basis functions learned from solving {Tk}M
+k=1 with MH-PINNs, and
+the density of the head, estimated from its samples using NFs, acts as the regularization, the prior
+distribution, or the generator together with the body, depending on the usage of MH-PINNs in ap-
+plications.
+2.1. Multi-head physics-informed neural networks (MH-PINNs). Hard
+parameter sharing is the most commonly used approach when MTL with NNs are
+considered, and MH-NNs, as its simplest instance, are frequently adopted [5, 39].
+A MH-PINN, as described earlier, is composed of a body and multiple heads. We
+denote by Φ the body and by Hk the head for Tk. Notice that here Φ : RDx → RL is
+a function parameterized by a neural network with parameter θ, and Hk ∈ RL+1 is a
+vector, where L is the number of neurons on the last layer of the body. Let us define
+Hk = [h0
+k, h1
+k, ..., hL
+k ]T , Φ(x) = [φ1(x), ..., φL(x)]T , where φ : RDx → R, and then the
+surrogate for the solution in Tk can be rewritten as ˆuk(x) = h0
+k+�L
+l=1 hl
+kφl(x), ∀x ∈ Ω.
+The approximated source terms and boundary/initial terms are derived from Eq. (2.1)
+accordingly. In the MTL framework , given data {Dk}M
+k=1 and physics Eq. (2.1), the
+loss function L is formulated as follows:
+(2.2)
+L({Dk}M
+k=1; θ, {Hk}M
+k=1) = 1
+M
+M
+�
+k=1
+Lk(Dk; θ, Hk),
+where Lk denotes the common loss function in PINNs. Conventionally, the data for
+Tk is expressed as Dk = {Df
+k, Db
+k, Du
+k}, where Df
+k = {xi
+k, f i
+k}
+N f
+k
+i=1, Db
+k = {xi
+k, bi
+k}N b
+k
+i=1 and
+
+samples of head
+Learned distribution of head
+1.2
+1.2
+1.0
+1.0 -
+0.8 
+0.8 
+learned by normalizing
+0.6
+0.6 
+flows
+0.4
+0.4 -
+0.2 
+0.2 
+0.0
+0.0 
+-1.0 
+-0.5
+0.0
+0.5
+1.0
+1.5
+1.0
+-0.5 
+0.0
+0.5
+1.0
+generative modeling { Hkl
+prior knowledge
+new tasks T
+Fine-tuning
+Body
+Bayesian inference
+1.00-0.750.500.250.000.250.500.751.00L-HYDRA
+5
+Du
+k = {xi
+k, ui
+k}N u
+k
+i=1, and Lk as follows:
+(2.3)
+Lk(Dk; θ, Hk) = wf
+k
+N f
+k
+N f
+k
+�
+i=1
+||Fk(ˆuk(xi
+k)) − f i
+k||2 + wb
+k
+N b
+k
+N b
+k
+�
+i=1
+||B(ˆuk(xi
+k)) − bi
+k||2
++ wu
+k
+N u
+k
+N u
+k
+�
+i=1
+||ˆuk(xi
+k) − ui
+k||2 + R(θ, Hk),
+where || · || represents a properly chosen norm, R(·) is a regularization method over
+the parameters of NNs, N f
+k , N b
+k, N u
+k are the numbers of data points for fk, bk, uk, and
+wf
+k, wb
+k, wu
+k are weights to balance different terms in the loss function.
+2.2. Generative modeling and normalizing flows (NFs). As mentioned
+earlier, MH-PINNs connect {Tk}M
+k=1 by making two assumptions: (1) the solutions
+uk, k = 1, ..., M share the same set of basis functions, Φ; and (2) the corresponding
+coefficients are samples of the same random variable, H. In [7], Φ was used as a carrier
+of prior knowledge from {Tk}M
+k=1 in downstream physics-informed learning tasks. In
+this work, we extend it by utilizing the information from the head as well by estimating
+the PDF and a generator of H from its samples, {Hk}M
+k=1, using normalizing flows
+(NFs). The interested readers are directed to [32, 22] for reviews of NFs as well as
+[9, 33, 20] for developments of some popular NFs.
+We choose NFs over other commonly used generative models, e.g., generative
+adversarial networks (GANs) [13], variational auto-encoders (VAEs) [21], or diffusion
+models [16], because the NF serves as both a density estimator and a generator. The
+former is able to provide proper regularization in the downstream few-shot physics-
+informed learning tasks, while the latter leads to a physics-informed generative method
+for approximating stochastic processes. It is worth noting that in previous works on
+physics-informed generative methods [44, 46, 15], NNs are trained by measurements
+over uk, and/or fk, and/or bk. Our model, on the other hand, learns through samples
+of the head, which is obtained from MTL in the first step. This learning strategy
+brings two substantial advantages: (1) flexibility in dealing with unstructured data,
+e.g., inconsistent measurements across tasks; (2) simplicity and controlability of the
+training by decoupling the physics-informed learning and the generative modeling.
+2.3. Prior knowledge utilized in the downstream tasks. Here, we describe
+details on how to utilize the prior knowledge stored in MH-PINNs, for downstream
+few-shot physics-informed learning task, ˜T , which is defined the same as all other
+tasks in the upstream training, but with much fewer measurements. Training of MH-
+PINNs and NFs yield a body, Φ, samples of heads, {Hk}M
+k=1, and an estimated PDF
+of the head, ˆp(H) ≈ p(H). In solving ˜T with ˜D, we fix the body Φ and find the head
+˜H that best explains the data ˜D and the physics in Eq. (2.1). Noiseless and noisy data
+are considered in this paper: for noiseless data, regular NN training is performed on
+the head for new tasks to provide deterministic predictions, where the learned PDF
+of the head, ˆp(H), acts as a regularization term in the loss function; for noisy data,
+Bayesian inference is performed on the head as well, in which ˆp(H) denotes the prior
+distribution. Details are presented in the following.
+2.3.1. Regularization in optimization. Limited data in few-shot learning
+often leads to over-fitting and/or poor inter-/extrapolation performance. In this re-
+gard, regularizing the head according to its PDF is able to prevent over-fitting and
+
+6
+Z. ZOU AND G. E. KARNIADAKIS
+provide additional prior knowledge for better inter-/extrapolation performance. The
+optimization problem is cast as
+(2.4)
+˜H∗ = arg min
+˜
+H
+L∗( ˜D; ˜H), where L∗( ˜D; ˜H) = L( ˜D; ˜H) − α log p( ˜H)
+≈ L( ˜D; ˜H) − α log ˆp( ˜H),
+where L is the regular loss function in physics-informed learning for data ˜D and param-
+eter ˜H, and α ≥ 0 is the coefficient to adjust the regularization effect. Problem (2.4)
+in this work is solved with gradient descent.
+2.3.2. Prior distribution in Bayesian inference. As opposed to point esti-
+mate obtained by solving the optimization problem (2.4), the posterior distribution
+of the head in ˜T is obtained using Bayesian inference. Similar as in [30, 47], the
+posterior distribution of ˜H is established as follows:
+(2.5)
+p( ˜H| ˜D) ∝ p( ˜D| ˜H)p( ˜H) ≈ p( ˜D| ˜H)ˆp( ˜H),
+where p( ˜H| ˜D) is the posterior distribution, p( ˜D| ˜H) is the likelihood distribution,
+which is often assumed to be independent Gaussian over all measurements in ˜D,
+and ˆp is the estimated PDF of the head via NFs. Intractability of distribution (2.5)
+requires approximation methods, among which Markov chain Monte Carlo methods,
+such as Hamiltonian Monte Carlo (HMC) [31], generally provide the most accurate
+estimation.
+Moreover, the relatively low dimension of ˜H also enables the use of
+Laplace’s approximation (LA) [18], which is employed in this paper as an alternative
+to HMC.
+3. Related works. Deep NNs in recent years have been extensively investigated
+for solutions of ODEs/PDEs, SDEs as well as operator learning. Although not explic-
+itly introduced as MH-PINNs, MH-NNs were first used to solve ODEs/PDEs by [7],
+in which MH-PINNs were pre-trained on multiple similar tasks, and then the heads
+were discarded while the body was kept and transferred to solving new tasks, by either
+least square estimate for linear ODEs/PDEs, or fine-tuning with gradient descent for
+nonlinear ones. In [7], a one-shot transfer learning algorithm for linear problems was
+proposed but other potential uses of MH-NNs, e.g., MTL and generative modeling,
+were not discussed, as opposed to the work presented herein. Furthermore, [7] focused
+only on fast and deterministic predictions with high accuracy using sufficient clean
+data, while in this paper, we study the applicability of MH-NNs to few-shot physics-
+informed learning as well, where data is insufficient and/or noisy, and address such
+cases with UQ. We note that MH-NN was also used as a multi-output NN in [45],
+which, however, focused on solving single tasks and obtaining uncertainties.
+Generative modeling in the context of scientific computing has also been studied
+recently, and a few attempts for adopting deep generative NNs to SciML problems
+have been made in [44, 46, 15], most of which have focused on approximating stochas-
+tic processes and on solving SDEs. We propose a new physics-informed generative
+method, as an alternative to the current ones, using MH-PINNs, and test it in approxi-
+mating stochastic processes. In this regard, our method is functionally the same as the
+current ones, but technically different. All previous methods address physics-informed
+generative modeling using end-to-end learning strategies by coupling two dissimilar
+types of learning, physics-informed learning and generative modeling, which may be
+problematic for implementation and usage in practice when either type of learning
+becomes more complicated. Our method, on the other hand, addresses the problem in
+
+L-HYDRA
+7
+an entirely new angle by decoupling those two: physics-informed learning is performed
+first and is followed by learning generators. To this end, our method is a two-step
+method, and with the help of well-developed algorithms from both fields, our method
+has advantages both in flexibility and simplicity in implementation.
+4. Results. In this section, we test our method using five benchmarks. The first
+one is a pedagogical function regression, in which we aim to demonstrate the basic
+applicability and capabilities of our method, showing the importance of incorporating
+the distribution of the head in the downstream tasks and in obtaining results with
+or without uncertainty. The second example is a nonlinear ODE system, in which
+we test our method in approximating stochastic processes through a differential op-
+erator, compare different NFs, and eventually compare our method with another
+well-known physics-informed generative model, physics-informed GANs (PI-GANs)
+[44] in generative modeling. The third is a 1-D nonlinear reaction-diffusion equation,
+the fourth is a 2-D nonlinear Allen-Cahn equation, and the fifth is the 2-D stochastic
+Helmholtz equation with 20 dimensions. In all examples unless stated otherwise, data
+for {Dk}M
+k=1 are noise-free and task-wisely sufficient, while ˜D in downstream tasks is
+insufficient, which makes the downstream tasks of the few-shot type. In addition, ex-
+cept for the first example, results from Bayesian inference are obtained by employing
+HMC, and the predicted mean denoted as µ and predicted standard deviation denoted
+as σ are computed from the posterior samples of functions or unknown parameters.
+The predicted uncertainty is defined as 2σ in this paper.
+4.1. Function approximation. We start with a function regression problem
+using only data and no physics, which is a degenerate instance of Eq. (2.1) with Fk
+being fixed as an identity operator, no B and bk, and uk = fk being task-specific. In
+this case, Dk and ˜D are given as {(xi
+k, f i
+k)}Nk
+i=1 and {(xi, f i)}N
+i=1, respectively, and Tk
+and ˜T are defined as approximating functions fk and ˜f from Dk and ˜D, respectively.
+The stochastic function f in this example is defined as follows:
+(4.1)
+f(x) = A cos(ωx) + 2βx, x ∈ [−1, 1],
+A ∼ U[1, 3), ω ∼ U[2π, 4π), P(β = ±1) = 0.5,
+where U stands for uniform distribution and P(Ξ) is defined as the probability of
+the event Ξ. Our goal is to approximate f from data with MH-NNs and NFs, and
+solving the downstream few-shot regression tasks ˜T as well, in which two functions,
+2 cos(2πx)+2x and 2 cos(4πx)−2x, are regressed from 4 and 5 measurements equidis-
+tantly distributed on [−0.9, −0.1], respectively.
+For the training of MH-NNs and NFs, 1, 000 f subject to Eq. (4.1) are sampled,
+each of which forms a regression task with 40 measurements sampled equidistantly on
+[−1, 1] as data. Samples of f for training are displayed in Fig. 3(a). Both noiseless
+and noisy data cases are considered in the few-shot regression tasks. As described in
+Sec. 2.3, the former is solved by fine-tuning the head using gradient descent, while
+the latter is solved by estimating the posterior distribution (Eq. (2.5)) using HMC
+and LA. The noise ε is assumed to be independent additive Gaussian noise with scale
+0.2, i.e., ε ∼ N(0, 0.22). In the downstream few-shot regression tasks we compare our
+method with two other approaches, the transfer learning (TL) method from [7], which
+only transfers the body, Φ, and the regular NN method, in which no prior knowledge
+is employed and all parameters of NN are trained.
+Results for approximating f and solving the downstream tasks are presented in
+Fig. 3. As shown in Fig. 3(a), our method approximates the stochastic function f
+
+8
+Z. ZOU AND G. E. KARNIADAKIS
+(a)
+(b)
+(c)
+Fig. 3. Results for approximating the stochastic function defined in Eq. (4.1) and solving the
+downstream few-shot regression tasks. (a) Left: 1, 000 samples generated from the exact distribu-
+tion; middle: 1, 000 samples generated from the learned generator; right: statistics computed from
+samples, in which we refer to the interval of mean ± 2 standard deviations as bound. (b)/(c) Results
+for the downstream tasks. Left: results for noiseless cases using our method, the transfer learning
+(TL) method in [7], and regular NN method; middle: results for noisy case using our method with
+HMC for posterior estimate; right: results for the same noisy case using our method with LA for
+posterior estimate.
+well, demonstrating the capability of MH-NNs in generative modeling. In solving the
+downstream tasks with noiseless data, L2 regularization is imposed in the TL method
+and regular NN method, to prevent over-fitting when only 4 or 5 measurements are
+available. As we can see from Figs. 3(b) and (c), our approach yields accurate predic-
+tions and performs significantly better than the other two in both tasks, particularly
+in the region where there are no measurements. By comparing our approach with
+the NN method, we can see that prior knowledge of f is learned from {Tk}M
+k=1 and
+transferred successfully to new downstream tasks. By comparing our approach with
+the TL method, we can see that the prior knowledge is stored in both the body and
+(the distribution of) the head. For the noisy cases, it is shown that, for both tasks and
+both posterior estimating methods, the predictions are accurate and trustworthy: the
+predicted means agree with the references and the errors are bounded by the predicted
+uncertainties. It is worth noting that the predicted uncertainties do not develop in the
+interval [0, 1] and show periodic patterns, even if there are no measurements. That is
+because an informative prior, which is learned by MH-NNs and NFs, is imposed on
+the head in Bayesian inference.
+The target functions in downstream tasks considered previously are chosen to be
+in-distribution. They are regressed well with insufficient data, mainly because they
+
+4
+3
+2
+0
+-1
+-2
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.84
+3
+2
+0
+-1
+-2
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.86
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.86
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8Predicted mean
+Predicted bound
+Reference mean
+Reference bound
+2
+.2
+6
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+Measurements
+Reference
+Ours
+3
+TL
+2
+NN
+0
+1
+-2 
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+2 std
+4
+Measurements
+Reference
+3
+Mean
+2
+0
+-1
+-2 
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+2 std
+4
+Measurements
+Reference
+3
+Mean
+2
+0
+-1
+-2 
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+4
+0
+-1
+-2 
+-3
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8L-HYDRA
+9
+belong to the space of functions, on which the generator is trained. However, when
+functions in the downstream tasks are out-of-distribution (OOD), our approach fails
+to produce good predictions, even if the data is sufficient, as shown in Fig. 4. Here,
+the target function is chosen to be 2 cos(4.5π) + x with both ω and β being OOD.
+Fluctuations are predicted but do not match the reference. In Fig. 4, we can further
+see that when data is sufficient, a NN trained from scratch significantly outperforms
+our approach, showing that, for OOD functions, the more we rely on the learned
+regularization, which is indicated by the value of α in Eq. (2.4), the more erroneous
+the prediction is.
+Fig. 4. Results for regression on an out-of-distribution function. Left: few-shot regression with
+clean data using our approach; middle: few-shot regression with noisy data using our approach with
+HMC for posterior estimate; right: regression with sufficient clean data using regular NN method
+and our approach with different regularization terms, α in Eq. (2.4).
+4.2. Nonlinear ODE system. In this example, we consider the following ODE
+system [28], which describes the motion of a pendulum with an external force:
+(4.2)
+du1
+dt = u2,
+du2
+dt = −λ sin(u1) + f(t),
+with initial condition u1(0) = u2(0) = 0. In Eq. (4.2), f is the external force and
+λ is a constant. Here, to demonstrate and study the capability of our method in
+generative modeling, we first consider the case where f is a Gaussian process and
+λ = 1 is known, which is referred to as the forward problem. Different from previous
+studies [44, 46, 15], in which a stochastic process is approximated directly by the
+output of NNs, in this example we place the differential operator right after NNs and
+approximate the stochastic process f as the source term in Eq. (2.1). We also test
+our method on the inverse problem, where the values of λ in Eq. (4.2) are unknown in
+{Tk}M
+k=1 and ˜T . The forward problem corresponds to Eq. (2.1) with Fk, bk being the
+same for all tasks and uk, fk being task-specific, while the inverse problem corresponds
+to Eq. (2.1) with bk being the same and uk, fk, and the differential operator Fk being
+different as a consequence of task-specific λ.
+4.2.1. Forward problem. We first assume λ = 1 in Eq. (4.2) is known, and
+the data on f is available, i.e., Dk = {(xi
+k, f i
+k)}Nk
+i=1, k = 1, ..., M. As described before,
+we employ MH-PINNs to solve {Tk}M
+k=1 all at once and then employ NFs to learn the
+distribution of the head. Consequently, we obtain generators of f and u. In this case,
+f is assumed to be a Gaussian process with squared kernel function:
+(4.3)
+f(t) ∼ GP(0, K), t ∈ [0, 1], K(x, x′) = exp(−|x − x′|2
+2l2
+),
+where the correlation length l is set to 0.1, 0.25, 0.5. As discussed in Sec. 2.2, many
+types of NFs have been developed in the past decade for generative modeling and
+
+5
+Measurements
+Reference
+Ours
+3
+0
+-2
+-3
+-0.8
+-0.6
+-0.4
+-1
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+2 std
+4
+Measurements
+Reference
+3
+Mean
+2
+-2 
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+Measurements
+4
+Reference
+NN
+3
+= 10-6
+2
+= 10-4
+10-2
+0
+-2
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+110
+Z. ZOU AND G. E. KARNIADAKIS
+density estimate. MH-PINNs, when used as generators, are compatible with all NFs.
+In this regard, we compare three popular NFs, RealNVP [9], MAF [33] and IAF [20],
+and eventually compare MH-PINNs (with NFs) against PI-GAN [44] in approximating
+the Gaussian process defined in Eq. (4.3) with different correlation lengths.
+For the training of MH-PINNs and NFs as well as PI-GANs, 2, 000 f are sampled
+with respect to Eq. (4.3), each of which forms a physics-informed regression task with
+65 measurements of f equidistantly sampled on [0, 1] as data. Notice that the ODE
+system in Eq. (4.2) can be rewritten in a simpler format as follows:
+(4.4)
+utt = −λ sin(u) + f(t), t ∈ [0, 1],
+with initial conditions u(0) = ut(0) = 0. Hence, we choose to use Eq. (4.4) to build
+the loss function for physics-informed learning.
+Results for comparisons are shown in Fig. 5 and Table 1.
+From the spectral
+analysis of the approximated Gaussian processes shown in Fig. 5, we can see that
+MH-PINNs with MAF and IAF are comparable with PI-GANs while MH-PINNs
+with RealNVP fall marginally behind. As shown in Table 1, the computational costs
+of MH-PINNs with MAF and RealNVP are significantly lower than PI-GANs, while
+MH-PINNs with IAF is more expensive than PI-GANs. We note that in this example
+we also record the computational cost for sampling using different generators. As
+shown in Table 1, PI-GANs are significantly faster in generating samples. That is
+because, generally, GANs require relatively shallow NNs as opposite to NFs, for which
+a deep architecture is needed to keep up the expressivity. Among three NFs, IAF is
+the fastest in sampling while MAF is the lowest, as opposed to training, which is
+consistent with the properties of those two NFs: MAF is slow for the forward pass,
+which is used to generate samples, and fast for the inverse pass, which is used to
+compute the density, while IAF is the opposite. Despite the fact that MAF is slow in
+sampling, considering its fast training and good performance, we equip MH-PINNs
+with MAF as the density estimator and the generator for all other examples in this
+paper.
+Fig. 5.
+Approximating Gaussian processes as the source term in Eq. (4.2) using different
+models:
+spectra of the correlation structure for the learned generators, for different correlation
+lengths, l. The covariance matrix is constructed using 10, 000 generated samples, and eigen-values
+are averaged over 10 generators trained independently.
+4.2.2. Inverse problem. Next, we assume λ in Eq. (4.2) is unknown, and some
+measurements of u are available, in addition to f, i.e., Dk = {{xi
+k, f i
+k}
+N f
+k
+i=1, {xi
+k, ui
+k}N u
+k
+i=1}.
+MH-PINNs are first employed to infer uk as well as λk from data Dk and physics, and
+NFs are employed afterwards to learn from samples of Hk and λk. To this end, the
+generative model is for the joint distribution of u, f and λ. Here, we assume f follows
+a truncated Karhuen-Loeve (KL)-expansion, with 5 leading terms, of the Gaussian
+process with squared kernel function and correlation length 0.1, and for each task Tk,
+
+1 = 0.1
+0.3
+e-Reference
+PI-GAN
+0.25
+MAF
+IAF
+0.2
+RealNVP
+eigenvalues
+0.15
+0.1
+0.05
+0
+0
+5
+10
+15
+components1 = 0.25
+0.7
+0.6
+0.5
+eigenvalues
+0.4
+0.3
+0.2
+0.1
+0
+0
+5
+10
+15
+components1 = 0.5
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+5
+10
+15
+componentsL-HYDRA
+11
+MAF
+IAF
+RealNVP
+PI-GAN
+Phase 1
+134s
+134s
+134s
+N/A
+Phase 2
+252s
+3939s
+245s
+N/A
+Total
+386s
+4073s
+379s
+3243s
+Sampling
+1.98 × 10−1s
+1.48 × 10−2s
+1.50 × 10−2s
+2.29 × 10−3s
+Table 1
+Computational time for different models to approximate Gaussian process with correlation
+length l = 0.1.
+The MH-PINN method is a two-step method and hence its computation is de-
+composed into two parts: training MH-PINNs referred to as phase 1 and training NFs referred to
+as phase 2. Sampling time is defined to be the average time needed to generate 10, 000 samples of u.
+λk = 1
+2 exp(
+�
+[0,1] f 2
+k(t)dt). As for the downstream task ˜T , the target is to infer u and
+λ from insufficient data of u and f.
+For the training of MH-PINNs and NFs, 2, 000 samples of f are generated and
+displayed in Fig. 6(a). For each task, we assume 33 measurements of fk and 9 mea-
+surements of uk, equidistantly distributed on [0, 1], are available, and initial conditions
+are hard-encoded in NN modeling. For the downstream task, we assume 1 random
+measurement of u and 8 random measurements of f are available with hard-encoded
+initial conditions as well.
+For the case with noisy measurements, we assume the
+noises εf and εu to be additive Gaussian, with 0.05 noise scale for measurements of
+f and 0.005 noise scale for measurements of u, respectively, i.e. εf ∼ N(0, 0.052)
+and εu ∼ N(0, 0.0052). The reference solution as well as the clean data of uk are
+generated by solving Eq. (4.2) for each task Tk, with corresponding fk and λk using
+Matlab ode45.
+Results are shown in Fig. 6 and Table 2, from which we can see our method is able
+to approximate the stochastic process as a source term well and produce accurate and
+trustworthy predictions, for u, f and also λ in the downstream task with limited data,
+in both noiseless and noisy cases. As shown, the PINN method yields unacceptable
+estimate over both u and λ due to lack of data, while our approach is of much higher
+accuracy by integrating prior knowledge from {Tk}M
+k=1 with MH-PINNs.
+PINN
+MH-PINN
+λ
+0.8440
+2.5428
+Error (%)
+63.99
+1.21
+Table 2
+Estimate of λ and L2 relative error of u for the downstream inverse problem on Eq. (4.2) with
+clean data, using our approach and the regular PINN method. The reference value for λ is 2.3609.
+4.3. 1-D nonlinear reaction-diffusion equation. We now test our method
+on a 1-D nonlinear time-dependent reaction-diffusion equation, which is commonly
+referred to as Fisher’s equation [2]:
+ut = Duxx + ku(1 − u), t ∈ [0, 1], x ∈ [−1, 1],
+(4.5)
+u(t, −1) = u(t, 1) = 0, t ∈ [0, 1],
+(4.6)
+u(0, x) = u0(x), x ∈ [−1, 1],
+(4.7)
+where D = 0.1, k = 0.1 and u0(x) is the initial condition function. In this example,
+we assume that the initial condition function is a stochastic process with the following
+
+12
+Z. ZOU AND G. E. KARNIADAKIS
+(a)
+(b)
+(c)
+Fig. 6. Results for the inverse problem of the ODE system (4.2), with initial conditions hard-
+encoded in NN modeling. (a) Left: 1, 000 samples of f, generated from the exact distribution; middle:
+1, 000 samples of f, generated from the learned generator; right: statistics computed from samples.
+The bound is defined as the same as in the caption of Fig. 3. (b) Predicted f and u using PINNs
+and our approach, for the downstream inverse problem with noiseless data. (c) Predicted f, u and
+λ with uncertainties using our approach, for the downstream inverse problem with noisy data. The
+predicted mean and standard deviation of λ is 2.4663 and 0.1501, while the reference value is 2.3609.
+distribution:
+(4.8)
+u0(x) = (x2 − 1)
+5
+5
+�
+j=1
+ξj(cos2(jx) − 1), x ∈ [−1, 1],
+where ξj, j = 1, ..., 5 are independent and identically distributed (i.i.d.) random vari-
+ables subject to uniform distribution on [0, 1), i.e., ξj ∼ U[0, 1).
+Unlike previous
+examples, the stochasticity comes from the initial condition rather than the source
+term. This example corresponds to Eq. (2.1) with Fk and fk being the same for all
+tasks, and uk and bk being task-specific. In addition to measurements of u0, points
+on which the PDE residuals are computed are also required in both {Tk}M
+k=1 and ˜T .
+Hence, the data is Dk = {{(ti
+k, xi
+k), 0}
+N f
+k
+i=1, {(0, xi
+k), bi
+k}N b
+k
+i=1}.
+For the training, 2, 000 samples of u0(x) are generated, displayed in Fig. 7(a),
+and each sample forms a physics-informed regression task with 41 measurements of
+u0 equidistantly sampled on [−1, 1] as data for initial condition. Besides, for all tasks,
+a uniform mesh 21 × 41 on temporal-spatial domain [0, 1] × [−1, 1] is used to com-
+pute the PDE residual loss. For the downstream tasks ˜T , 5 random measurements
+of u0 are available and the same uniform mesh is applied. The boundary conditions
+are hard-encoded in NN modeling in both {Tk}M
+k=1 and ˜T . For the noisy case, the
+
+4
+3
+2
+-3
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.94
+3
+2
+-3
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.94
+Predicted mean
+Predicted bound
+3
+Reference mean
+Reference bound
+2
+-3
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+t2
+Measurements
+1.5
+Reference
+ Ours
+1
+..- PINN
+0.5
+0
+-0.5
+-1
+-1.5
+-2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+t0.15
+0.1
+U
+0.05
+0
+-0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+t2
+2 std
+1.5
+Measurements
+Reference
+Mean
+0.5
+0
+-0.5
+-1
+-1.5
+-2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+t0.15
+0.1
+0.05
+0
+-0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+t2.5
+Prior (learned from physics)
+Posterior
+2
+1.5
+0.5
+2
+3
+4
+5
+6L-HYDRA
+13
+noise ε is assumed to be independent additive Gaussian noise with 0.02 noise scale
+for both measurements of u0 and the PDE residual, i.e., ε ∼ N(0, 0.022). Results are
+presented in Fig. 7 and Table 3. We can see that our method estimates a good gen-
+erator of the stochastic processes from data and physics, which provides informative
+prior knowledge in the downstream few-shot physics-informed regression tasks. The
+prediction is accurate in both noiseless and noisy cases, and the errors in the noisy
+case are bounded by the predicted uncertainty. The L2 error of u, shown in Table 3,
+indicates that our approach outperforms the PINN method by a significant amount,
+hence demonstrating the effectiveness of bringing prior knowledge into solving similar
+tasks.
+(a)
+(b)
+(c)
+Fig. 7.
+Generator learning and few-shot physics-informed learning on 1-D time-dependent
+reaction-diffusion equation (4.5), with boundary conditions hard-encoded in NN modeling. (a) Left:
+1, 000 training samples of u0; middle: 1, 000 samples of u(0, ·) from the learned generator; right:
+statistics computed from samples. The bound is defined as the same as in the caption of Fig. 3. (b)
+Predicted u at t = 0, 0.5, 1 using our approach and the PINN method with noiseless measurements.
+(c) Predicted mean and uncertainty of u at t = 0, 0.5, 1 using our approach with HMC for posterior
+estimate, with noisy measurements.
+PINN
+MH-PINN
+Error (%)
+78.77
+0.22
+Table 3
+L2 relative error of u for the downstream few-shot physics-informed learning task on Eq. (4.5)
+with clean data of u0 using our approach and the PINN method.
+
+0.5
+0.4
+0.3
+0=4
+0.2
+u
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+0.4
+0.3
+0=4
+0.2
+u
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+Predicted mean
+Predicted bound
+Reference mean
+0.4
+Reference bound
+0.3
+0=1
+0.2
+0.1
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+Measurements
+Reference
+0.4
+Ours
+..-PINN
+0.3
+0=1
+0.2
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+0.4
+0.3
+t=0.5
+0.2
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+0.4
+0.3
+t=1
+0.2
+u
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+2 std
+Measurements
+0.4
+Reference
+Mean
+0.3
+0=1
+0.2
+f
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+0.4
+0.3
+t=0.5
+0.2
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.5
+0.4
+0.3
+t=1
+0.2
+u
+0.1
+0
+-0.1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.814
+Z. ZOU AND G. E. KARNIADAKIS
+4.4. 2-D nonlinear Allen-Cahn equation. We now move to a 2-D steady
+nonlinear Allen-Cahn equation with Dirichlet boundary conditions [43]:
+λ∆u + u(u2 − 1) = f, x, y ∈ [0, 1],
+(4.9)
+u(x, 0) = u(x, 1) = u(0, y) = u(1, y) = 0,
+(4.10)
+where λ = 0.1 is a constant and f is the source term. Here, we impose a distribution
+to f, which is derived from Eq. (4.9) and the following distribution of the solution u:
+(4.11)
+u(x, y) = 1
+5
+5
+�
+j=1
+ξj
+sin(jπx) sin(jπy)
+j2π2
+, x, y ∈ [0, 1],
+with i.i.d. random variables ξj, j = 1, ..., 5, subject to uniform distribution, i.e. ξj ∼
+U[0, 1). In this example, we wish to use our method to learn generators of both u and f
+from data of f and physics in Eq. (4.9), and use it to solve the downstream task ˜T with
+insufficient data ˜D. This example corresponds to Eq. (2.1) with Fk, bk being the same
+among tasks and fk, uk being task-specific, and the data is Dk = {(xi
+k, yi
+k), f i
+k}
+N f
+k
+i=1.
+To train the MH-PINNs and NFs, we sample 2, 000 f from its distribution, each
+of which is resolved with a 51 × 51 uniform mesh on 2-D spatial domain [0, 1] × [0, 1].
+As for the downstream task, 100 random measurements of f on the uniform mesh are
+assumed to be available. The noise is assumed to be independent additive Gaussian
+noise with 0.05 noise scale. In both Tk and ˜T , the boundary conditions are hard-
+encoded in NN modeling. Results as well as the locations of the measurements are
+presented in Fig. 8 and Table 4.
+Similar to all previous examples, our approach
+delivers accurate and trustworthy predictions, showing that prior knowledge is learned
+and transferred well in both deterministic and Bayesian inferences.
+(a)
+(b)
+Fig. 8. Results for few-shot physics-informed learning on the 2-D nonlinear Allen-Cahn equa-
+tion Eq. (4.9) with noisy measurements of f. Predicted mean µ and standard deviation σ are com-
+puted over 1, 000 posterior samples from HMC. The absolute error is defined as the absolute value
+of difference between the reference and µ. Black crosses represent the locations of the measurements
+on f.
+
+reference of f
+0.9
+0.2
+0.8
+0.7
+0.1
+0.6
+9
+0.5
+0
+0.4
+-0.1
+0.3
+0.2
+-0.2
+0.1
+0
+-0.3
+0
+0.2
+0.4
+0.6
+0.8predicted mean of f
+1
+x
+X
+X
+X
+0.9
+X
+X
+0.2
+X
+0.8
+X
+XX
+X
++
+X
+X
+0.7
+XX
+X
+0.1
+X
+×
+X
+X
+X
+0.6
+X
+X
+X
+++
+X
+x
+X
+9
+0.5
+X
+0
+X
+X
+0.4
+X
+X
+-0.1
+X
+X
+X
+0.3
+× XX
+XX
+0.2×
+X
+XX
+-0.2
+X
+X
+0.1
+X
+X
+X
+X
+X
+X
+X
+0
+-0.3
+0
+0.2
+0.4
+0.6
+0.8
+1absolute error of j
+1
+0.05
+交
+X
+XX
+X
+XX
+0.9
+X
+0.045
+X
+X
+X
+0.8
+0.04
+X
+X
+X
+X
+X
+0.7
+XX
+0.035
+X
+X
+X
+X
+0.6
+X
+X
+0.03
+X
+X
+++
+X
+X
+9
+0.5
+X
+0.025
+X
+X
+X
+X
+0.4
+XX
+X
+X
+X
+X
+0.02
+X
+X
+X
+X
+0.3
+X
+0.015
+XX
+XX
+XX ×
+X
+XX
+X
+0.2×
+0.01
+X
+XX
+X
+X
+0.1
+X
+X
+0.005
+X
+X
+X
+X
+X
++
+0
+0
+0
+0.2
+0.4
+0.6
+0.8
+1predicted uncertainty of f
+0.05
+0.9
+0.045
+0.8
+0.04
+0.7
+0.035
+0.6
+0.03
+9
+0.5
+0.025
+0.4
+0.02
+0.3
+0.015
+0.2
+0.01
+0.1
+0.005
+0
+0
+0
+0.2
+0.4
+0.6
+0.8reference of u
+0.01
+0.9
+0
+0.8
+0.7
+-0.01
+0.6
+9
+0.5
+-0.02
+0.4
+-0.03
+0.3
+0.2
+-0.04
+0.1
+0
+-0.05
+0
+0.2
+0.4
+0.6
+0.8
+1predicted mean of u
+0.01
+0.9
+0
+0.8
+0.7
+-0.01
+0.6
+9
+0.5
+-0.02
+0.4
+-0.03
+0.3
+0.2
+-0.04
+0.1
+0
+-0.05
+0
+0.2
+0.4
+0.6
+0.8
+1absolute error of u
+0.01
+0.9
+0.009
+0.8
+0.008
+0.7
+0.007
+0.6
+0.006
+9
+0.5
+0.005
+0.4
+0.004
+0.3
+0.003
+0.2
+0.002
+0.1
+0.001
+0
+0
+0
+0.2
+0.4
+0.6
+0.8
+1predicted uncertainty of u
+0.01
+0.9
+0.009
+0.8
+0.008
+0.7
+0.007
+0.6
+0.006
+9
+0.5
+0.005
+0.4
+0.004
+0.3
+0.003
+0.2
+0.002
+0.1
+0.001
+0
+0
+0
+0.2
+0.4
+0.6
+0.8
+1L-HYDRA
+15
+PINN
+MH-PINN
+Error (%)
+12.82
+0.30
+Table 4
+L2 relative error of u for the downstream few-shot physics-informed learning task on Eq. (4.9)
+with clean data of f, using our approach and the PINN method.
+4.5. 2-D Stochastic Helmholtz equation. The last example we test in this
+paper is the 2-D Helmholtz equation with stochastic source term and Dirichlet bound-
+ary conditions [35]:
+(λ2 − ∇2)u = f, x, y ∈ [0, 2π],
+(4.12)
+u(x, 0) = u(x, 2π) = u(0, y) = u(2π, y) = 0,
+(4.13)
+where λ2 is the Helmholtz constant and f is defined as follows:
+(4.14)
+f(x, y) = 2
+d{
+d/4
+�
+i=1
+ξi sin(ix) + ξi+d cos(ix) + ξi+2d sin(iy) + ξi+3d cos(iy)},
+where ξj, j = 1, ..., d are i.i.d. random variables subject to uniform distribution U[0, 1)
+and d represents the dimension of the randomness. For demonstration purposes, we
+consider the case where d = 20 in this paper, unlike the one in [35] with d = 100.
+The first case we study is the forward problem with λ2 = 1 known. This setup
+corresponds to Eq. (2.1) with Fk, bk being shared among tasks and uk, fk being task-
+specific. Next, we study the inverse problem with unknown λ, where data on u and f
+are available, which corresponds to Eq. (2.1) with only bk being the same and uk, fk
+and operator Fk being task-specific. The downstream tasks are defined as the same
+as {Tk}M
+k=1 in both cases, but with fewer measurements.
+For both the forward and inverse problems, 10, 000 f are sampled from its distri-
+bution, and hence 10, 000 tasks are solved with MH-PINNs with boundary conditions
+hard-encoded in NN modeling. We display the samples of a slice of f in Fig. 9(a).
+For the forward problem, Dk only contains measurements of the source term fk, i.e.
+Dk = {{(xi
+k, yi
+k), f i
+k}
+N f
+k
+i=1}, while for the inverse problem Dk also contains measure-
+ments of the sought solution uk: Dk = {{(xi
+k, yi
+k), f i
+k}
+N f
+k
+i=1, {(xi
+k, yi
+k), ui
+k}N u
+k
+i=1}. For the
+training in the forward problem, each sample of f is resolved by a 50 × 50 uniform
+mesh on 2-D spatial domain (0, 2π)×(0, 2π) with boundary excluded. For the inverse
+problem, the same 10, 000 samples of f are used, but this time they are resolved with
+a 21 × 21 uniform mesh. In addition, for each task Tk, measurements of uk on a 6 × 6
+uniform mesh are available. The reference solution and measurements of u are gen-
+erated by solving Eq. (4.12) with λ2
+k =
+�
+[0,2π]2 f 2
+k(x, y)dxdy using the finite difference
+method with five-point stencil. For the downstream tasks, 100 random measurements
+of f are available for the forward problem, and 50 random measurements of f and
+10 random measurements of u are available for the inverse problem. The noise is
+assumed to be independent additive Gaussian noise with 0.05 noise scale.
+Results are displayed in Tables 5 and 6, and Figs. 9 and 10.
+As shown, the
+learned generator is able to produce samples of f with high quality as well as providing
+informative prior knowledge for the downstream tasks, in both the forward and inverse
+problems. As for the noisy case with Bayesian inference and UQ, the predicted means
+agree with the references and the absolute errors are mostly bounded by the predicted
+uncertainties. The effectiveness of our approach for few-shot physics-informed learning
+
+16
+Z. ZOU AND G. E. KARNIADAKIS
+and the applicability to both deterministic optimization and Bayesian inference have
+been consistently demonstrated in the past five examples.
+(a)
+(b)
+(c)
+Fig. 9. Generator learning and few-shot physics-informed learning on the stochastic Helmholtz
+equation (4.12). (a) Left: 1, 000 training samples of a slice of f at y = π; middle: 1, 000 samples of
+a slice of f at y = π from the learned generator; right: statistics computed from samples. (b)/(c)
+Results for the downstream forward problem with 100 random noisy measurements on f, using our
+approach with HMC. From left to right are reference, predicted mean, absolute error, and predicted
+uncertainty of f/u. Black crosses represent the locations of the measurements of f.
+PINN
+MH-PINN
+Error (%)
+21.14
+1.12
+Table 5
+L2 relative error of u for the downstream forward problem on Eq. (4.12) with clean data of f,
+using our approach and the PINN method.
+PINN
+MH-PINN
+λ
+1.9328
+1.0170
+Error (%)
+59.92
+2.58
+Table 6
+Estimate of λ and L2 relative error of u for the downstream inverse problem on Eq. (4.12) with
+clean data. The reference value of λ is 1.0042.
+5. Multi-task learning with multi-head neural networks. So far we have
+mostly focused on using MH-NNs together with NFs to estimate stochastic generators
+
+0.4
+0.2
+=/
+9.
+0
+U
+-0.2
+-0.4
+-0.6
+0
+2
+3
+4
+5
+60.4
+0.2
+=
+9.
+0
+-0.2
+-0.4
+-0.6
+0
+2
+3
+4
+5
+6Predicted mean
+Predicted bound
+Reference mean
+0.4
+Reference bound
+0.2
+-0.2 
+-0.4
+-0.6
+0
+2
+3
+4
+5
+6reference of f
+6
+5
+0.5
+4
+9
+3
+0
+2
+1
+-0.5
+2
+3
+4
+5
+6
+1predicted mean of f
+6
+X
+X
+X
+X
+X
+X
+XX
+5
+X
+X
+X
+X
+X
+X
+0.5
+4
+X
+X
+X
+X
+9
+X
+3
+X
+X
+X
+XX
+0
+2 ×
+X
+X
+1
+X
+X
+X
+-0.5
+2
+3
+4
+5
+6absolute error of t
+0.1
+6
+0.09
+XX
+5
+0.08
+0.07
+0.06
+9
+0.05
+3
+0.04
+2×
+0.03
++
+0.02
+0.01
+X
+0
+2
+3
+4
+5
+6predicted uncertainty of f
+0.1
+6
+0.09
+0.08
+0.07
+4
+0.06
+9
+0.05
+3
+0.04
+2
+0.03
+0.02
+1
+0.01
+0
+2
+3
+4
+5
+6reference of u
+6
+0.04
+5
+0.02
+0
+4
+-0.02
+9
+3
+-0.04
+2
+-0.06
+1
+-0.08
+-0.1
+1
+2
+3
+4
+5
+6predicted mean of u
+6
+0.04
+5
+0.02
+0
+4
+-0.02
+9
+3
+-0.04
+2
+-0.06
+1
+-0.08
+-0.1
+2
+3
+4
+5
+6
+1absolute error of u
+0.02
+6
+0.018
+5
+0.016
+0.014
+4
+0.012
+9
+0.01
+3
+0.008
+2
+0.006
+0.004
+1
+0.002
+0
+1
+2
+3
+4
+5
+6predicted uncertainty of u
+0.02
+6
+0.018
+5
+0.016
+0.014
+4
+0.012
+9
+0.01
+3
+0.008
+2
+0.006
+0.004
+1
+0.002
+0
+1
+2
+3
+4
+5
+6L-HYDRA
+17
+(a)
+(b)
+Fig. 10.
+Results for the downstream inverse problem on the stochastic Helmholtz equa-
+tion (4.12), with 50 random noisy measurements of f and 10 random noisy measurements of u.
+λ is estimated as 1.0785 ± 0.0307 in the format of predicted mean ± predicted standard deviation,
+while the reference value is 1.0042. (a)/(b) From left to right are reference, predicted mean, absolute
+error, and predicted uncertainty of f/u. Black crosses represent locations of the measurements of f
+or u.
+and learn informative prior knowledge from {Tk}M
+k=1.
+This was achieved by first
+training MH-NNs in a MTL fashion and then training NFs to estimate the PDF of the
+head. Intuitively, the capability of MH-NNs when trained in MTL in capturing shared
+information is the key to the success in generative modeling and few-shot learning.
+For physics-informed MTL with MH-PINNs, ODEs/PDEs are solved simultaneously,
+and assuming the solutions to share the same set of basis functions gives us samples
+of the set of coefficients, which enables the generative modeling, followed by few-shot
+learning, which is the whole point of the method proposed in this paper. However,
+the cost and/or the benefit of imposing the same set of basis functions to all solutions
+have not been explicitly discussed yet. On one hand, the shared body relates the
+training of tasks, which may be helpful if tasks are similar in certain ways. On the
+other hand, forcing all solutions to share the same basis functions may also be harmful
+when they behave differently. In particular, for tasks with sufficient data and physics,
+forcing them to share the same body with all other tasks may act as a negative
+regularization, and single-task learning (STL) may outperform MTL in terms of the
+prediction accuracy in those specific tasks. In this section, we investigate the effect
+of MTL using MH-NNs and provide preliminary results and analysis by revisiting the
+simple function approximation example in Sec. 4.1, which, hopefully, could provide
+useful information and insight for future more rigorous research.
+5.1. Basis function learning and synergistic learning. As discussed before,
+the quality and behavior of basis functions learned in MTL are crucial to genera-
+tive modeling and learning the relation and the representative information of tasks
+{Tk}M
+k=1. We consistently noticed from numerical examples that the initialization of
+the head in MH-NNs has great impact on the average accuracy of MTL, the learning of
+the basis functions, and the distribution of the head. Here, we test three initialization
+strategies, random normal method with 0.05 standard deviation referred to as RN
+
+reference of f
+6
+5
+0.5
+4
+9
+3
+0
+2
+1
+-0.5
+1
+2
+3
+4
+5
+6predicted mean of f
+x
+6
+X
+X
+X
+X
+X 
+ X
+X
+X
+5
+X
+X
+X
+X
+X
+X
+X
+0.5
+4
+X
+X
+X
+X
+X
+X
+X
+9
+3
+X
+X
+X
+X
+X 
+X
+0
+2
+X
+X
+X
+X
+X
+X
+X
+X
+1
+X
+X
+X
+-0.5
+1
+2
+3
+4
+5
+6absolute error of i
+0.1
+6
+0.09
+5
+0.08
+0.07
+4
+X
+0.06
+9
+0.05
+3
+X
+X
+ X
+0.04
+2
+0.03
+0.02
+0.01
+0
+2
+3
+4
+5
+6predicted uncertainty of f
+0.1
+6
+0.09
+5
+0.08
+0.07
+4
+0.06
+9
+0.05
+3
+0.04
+2
+0.03
+0.02
+0.01
+0
+2
+3
+4
+5
+6reference of u
+6
+0.04
+5
+0.02
+0
+4
+-0.02
+9
+3
+-0.04
+2
+-0.06
+1
+-0.08
+-0.1
+1
+2
+3
+4
+5
+6predicted mean of u
+6
+0.04
+X
+5
+0.02
+X
+X
+0
+4
+-0.02
+X
+X
+9
+3
+-0.04
+2
+X
+X
+X
+-0.06
+1
+-0.08
+X
+-0.1
+1
+2
+3
+4
+5
+6absolute error of u
+0.02
+6
+0.018
+5
+0.016
+0.014
+4
+0.012
+9
+x
+X
+0.01
+3
+0.008
+2
+X
+0.006
+0.004
+1
+X
+0.002
+0
+1
+2
+3
+4
+5
+6predicted uncertainty of u
+0.02
+6
+0.018
+5
+0.016
+0.014
+4
+0.012
+9
+0.01
+3
+0.008
+2
+0.006
+0.004
+1
+0.002
+0
+1
+2
+3
+4
+5
+618
+Z. ZOU AND G. E. KARNIADAKIS
+(0.05), Glorot uniform method [12] referred to as GU, and random normal method
+with 1 standard deviation referred to as RN (1). In the downstream few-shot learning
+tasks, we fine-tune the head without the learned PDF, which is in fact the TL method
+from [7], by which the information from the distribution of the head is excluded and
+the prediction accuracy is fully determined by the level of prior knowledge contained
+in the basis functions.
+As shown in Fig. 11, RN (0.05) yields the least informative basis functions, whose
+behavior is dominated by the hyperbolic tangent activation function of NNs. This is
+further demonstrated in the downstream few-shot learning tasks using the TL method.
+It also provides the worst prediction accuracy on average in MTL, as presented in
+Table 8. GN and RN (1) perform similarly. Plots of some basis functions seemingly
+indicate that RN (1) yields better basis functions, whose behaviors are more similar
+to the family of functions displayed in Fig. 11(b), which, however, does not necessarily
+imply richer prior knowledge in the downstream tasks, as shown in Fig. 11(c).
+It is shown empirically that compared to other two initialization strategies, MH-
+NNs with RN (0.05) does not deliver accurate MTL nor synergistic learning in basis
+functions. However, we noticed that, in generative modeling, it performs significantly
+better in terms of accuracy and convergence speed. As shown in Fig. 11(d), samples
+from the learned generator are of higher quality. We consistently found that initializ-
+ing heads with relatively small values often led to easy and fast training of NFs and
+accurate learning of the generative models. We conjecture that this happens because
+MH-NNs in MTL tend to contain the representative and informative information in
+the heads when heads are initialized with small values, while contain it in the basis
+functions when heads are initialized with relatively large values.
+RN (0.05)
+GU
+RN (1)
+Error (%)
+0.8373 ± 0.2341
+0.1907 ± 0.0690
+0.3131 ± 0.0937
+Table 7
+L2 relative errors, from MTL, for 1, 000 tasks, using different initialization methods.
+The
+errors are displayed in the format of mean ± standard deviation, computed over all tasks.
+5.2. Multi-Task Learning (MTL) versus Single-Task Learning (STL).
+As discussed earlier, MTL with MH-NNs does not necessarily result in synergistic
+learning nor higher accuracy for all tasks on average. Here, we use again the function
+approximation example in Sec. 4.1, to investigate the effectiveness of MTL with MH-
+NNs, as compared to STL. The first case we consider here is assuming that the data is
+sufficient. For that, we randomly choose 100 out of the 1, 000 training samples, each
+one of which is approximated by a NN trained independently, and compare the results
+with MH-NNs in terms of prediction accuracy. Note that in this case, a MH-NN is
+trained on 1, 000 functions as before and tested on the chosen 100 functions, while a
+single-head NN with the same architecture is trained on 100 functions directly. Results
+are shown in Table 8, from which it is verified empirically that MTL is outperformed
+by STL under certain circumstances, e.g., when the random normal initialization
+methods are used.
+The second case we consider is assuming that the data is sufficient for some tasks
+while insufficient for other tasks. For that, we split equally the 1, 000 tasks into two
+subsets of tasks. For the first 500 tasks, we assume we only have 10 measurements
+randomly sampled on [−1, 1], while for the other 500 tasks, we assume we have full 40
+measurements equidistantly distributed on [−1, 1]. MTL with MH-NNs is performed
+on those 1, 000 regression tasks all at once, and the tasks are treated as equal. The
+
+L-HYDRA
+19
+(a)
+(b)
+(c)
+(d)
+Fig. 11. The effect of different initialization methods of the head, in basis functions learning,
+few-shot learning, and generator learning. (a) Samples of 20 basis functions from MH-NNs, trained
+for approximating 1, 000 f generated from Eq. (4.1), using, from left to right, RN (0.05), GU and
+RN (1) initialization methods. (b) 1, 000 training samples of f. (c) Results for two downstream
+few-shot regression tasks, using TL method without regularization informed by the learned PDF, as
+opposite to the proposed approach. (d) Results for generator learning, using, from left to right, RN
+(0.05), GU and RN (1) initialization methods.
+RN (0.05)
+GU
+RN (1)
+STL
+Error (%)
+0.7575 ± 0.2477
+0.1362 ± 0.0259
+0.3664 ± 0.1031
+0.2102 ± 0.0794
+Table 8
+L2 relative errors of f, from MTL with MH-NNs and STL with NNs, on 100 tasks. Different
+initialization methods are used for the heads in MH-NNs. The errors are displayed in the format of
+mean ± standard deviation, computed over all 100 tasks.
+results are presented in Table 9 and Fig. 12. We can see that, compared to STL, MTL
+improves the prediction accuracy on tasks with insufficient data, providing empirical
+evidence of synergistic learning.
+Also, interestingly, RN (1) initialization method,
+which yields the worst generative models, performs the best among all three, which
+agrees with our previous conjecture on the basis functions learning with MH-NNs,
+that heads initialized with large values tend to force representative and informative
+information to be encoded in the basis functions.
+6. Discussion. We have developed multi-head neural networks (MH-NNs) for
+physics-informed machine learning, and proposed multi-head physics-informed neural
+networks (MH-PINNs) as a new method, implemented in the L-HYDRA code. The
+primary focus of this work is on MH-NNs and MH-PINNs for various learning prob-
+
+6
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.8
+0.6
+0.4
+0.2
+0
+-0.2
+-0.4
+-0.6
+-0.8
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.80.8
+0.6
+0.4
+0.2
+0
+-0.2
+-0.4
+-0.6
+-0.8
+1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.81
+0.8
+0.6
+0.4
+0.2
+0
+-0.2
+-0.4
+-0.6
+-0.8
+1
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8Measurements
+Reference
+RN (0.05)
+3
+GU
+2
+RN
+-2
+-3
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.84
+3
+2
+0
+-1
+-2
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.86
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.86
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.820
+Z. ZOU AND G. E. KARNIADAKIS
+RN (0.05)
+GU
+RN (1)
+Error (%)
+63.60 ± 24.08
+40.49 ± 20.49
+16.91 ± 11.08
+Table 9
+L2 relative errors of f, from MTL with MH-NNs, on 500 tasks equipped with insufficient data.
+The errors are displayed in the format of mean ± standard deviation, computed over all 500 tasks.
+Fig. 12. Results for 3 tasks with insufficient data from MTL with MH-NNs, using different
+initialization methods over the head, and from STL with NNs with the same architecture. We note
+that tasks with sufficient data and tasks with insufficient data are treated equally in MTL.
+lems in scientific machine learning, including multi-task learning (MTL), stochastic
+processes approximation, and few-shot regression learning. We first formulated the
+problem in Eq. (2.1), introduced the architecture design of MH-PINNs, and proposed
+a method to transform MH-NNs and MH-PINNs to generative models with the help
+of normalizing flows (NFs) for density estimation and generative modeling. We then
+studied the applicability and capabilities of MH-PINNs in solving ordinary/paritial
+differential equations (ODEs/PDEs) as well as approximating stochastic processes.
+We completed the paper with preliminary and empirical explorations of MH-NNs
+in synergistic learning, and examined the potential benefits and cost of MTL with
+MH-NNs.
+This paper can be used in various ways: it proposes a NN approach for MTL in
+solving ODEs/PDEs; it provides a new approach to approximate stochastic processes;
+it presents a method to address few-shot physics-informed learning problems, which
+are often encountered in the context of meta-learning and transfer learning; it contains
+a systematic study of applying MH-NNs to scientific computing problems; it presents
+the first empirical evidence of synergistic learning.
+However, there are a few major problems on MH-NNs we did not address, one
+of which is the expressivity of MH-NNs, or more generally hard-parameter sharing
+NNs in approximating complicated stochastic processes. Intuitively, if two functions
+behave very differently, forcing them to share the same basis functions would affect
+adversely the approximation accuracy. The second problem is the balancing issue of
+different terms in the loss function in MTL. It is shown in the literature [29] that
+PINNs, trained in single-task learning, are already deeply influenced by the weights
+in front of different terms in the loss function, e.g., data loss, boundary condition loss,
+PDE residual loss. This issue may be more complex in training MH-PINNs, because
+in MTL the loss function is commonly defined as weighted summation of task-specific
+loss. The last major problem is MH-PINNs for synergistic learning. In this paper, we
+only studied one example in function approximation and presented empirical evidence.
+More work for the understanding of synergistic learning with MH-PINNs along both
+the theoretical and computational directions should be pursued in the future.
+Acknowledgments. We would like to thank Professor Xuhui Meng of Huazhong
+University of Science and Technology for helpful discussions. This work was supported
+by: the Vannevar Bush Faculty Fellowship award (GEK) from ONR (N00014-22-
+
+Measurements
+Reference
+RN (0.05)
+3
+GU
+2
+RN(1)
+NN
+3
+-0.8
+-0.6
+-1
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+4
+3
+2
+0
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.85
+4
+3
+0
+-1
+-2 
+-3
+4
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8L-HYDRA
+21
+1-2795); the U.S. Department of Energy, Advanced Scientific Computing Research
+program, under the Scalable, Efficient and Accelerated Causal Reasoning Operators,
+Graphs and Spikes for Earth and Embedded Systems (SEA-CROGS) project, DE-
+SC0023191; and by the MURI/AFOSR FA9550-20-1-0358 project.
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+
+L-HYDRA
+23
+[45] M. Yang and J. T. Foster, Multi-output physics-informed neural networks for forward and
+inverse PDE problems with uncertainties, Computer Methods in Applied Mechanics and
+Engineering, (2022), p. 115041.
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+tic differential equations, arXiv preprint arXiv:2203.11363, (2022).
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+for uncertainty quantification in neural differential equations and operators, arXiv preprint
+arXiv:2208.11866, (2022).
+Appendix A. Details of NN architectures and training hyperparam-
+eters.
+For all examples in Secs. 4 and 5, MH-PINNs are implemented as fully-
+connected NNs (FNNs) with 3 nonlinear hidden layers, each of which is equipped
+with 50 neurons and hyperbolic tangent activation function. The number of heads is
+the same as the number of tasks in the corresponding examples: 1, 000 in Sec. 4.1,
+2, 000 in Secs. 4.2, 4.3 and 4.4, and 10, 000 in Sec. 4.5. Weights in the body of MH-
+PINNs are initialized with Glorot uniform initialization [12] and biases are initialized
+with zero, while heads are initialized by sampling from random normal distribution
+with 0.05 standard deviation, for fast training of NFs and better performance of the
+learned generators.
+Except for the forward problem in Sec. 4.2, NFs in this paper are chosen to be
+MAF [33] with 10 bijectors, i.e. the invertible map in NFs, each of which is a MADE
+[11], a NN with masked dense layers, with two nonlinear hidden layers equipped with
+100 neurons and ReLU activation function. The RealNVP [9] and IAF [20] used in
+the forward problem in Sec. 4.2 also have 10 bijectors, each of which is a NN with
+two nonlinear hidden layers equipped with 100 neurons and ReLU activation function.
+The implementation mostly follows the instructions of TensorFlow Probability library
+[8] for NFs.
+PI-GANs [44] implemented in Sec. 4.2 have the following architecture: the dis-
+criminator is a FNN with 3 nonlinear hidden layers, each of which is equipped with
+128 neurons and Leaky ReLU activation function; the generator that takes as input
+t is a FNN with 3 nonlinear hidden layers, each of which is equipped with 50 neu-
+rons and hyperbolic tangent activation function; the other generator takes as input a
+Gaussian random variable in 50 dimensions with zero mean and identity covariance
+matrix, and is implemented as a FNN with 3 nonlinear hidden layers, each of which
+has 128 neurons and hyperbolic tangent activation function. The input dimensions of
+those 3 FNNs are 65, 1 and 50, and the output dimensions are 1, 50, 50, respectively.
+For the training of MH-PINNs, full-batch training is deployed with Adam opti-
+mizer for 50, 000 iterations. For the training of NFs, except for the forward problem
+in Sec. 4.2, mini-batch training is deployed with batch size being 100 and Adam op-
+timizer for 1, 000 epochs.
+NFs in the forward problem in Sec. 4.2 are trained for
+500 epochs instead, and L2 regularization is imposed to the parameters of RealNVP
+for better performance. For all NFs, to achieve stable training, a hyperbolic tangent
+function is imposed on the logarithm of the scale, computed from each bijector, such
+that the logarithm of the scale lies in (−1, 1). For the training of PI-GANs, min-
+batch training is deployed with batch size being 100 and Adam optimizer for 100, 000
+iterations. Besides, the same as in [44, 30], physics-informed Wasserstein GANs (PI-
+WGANs) with gradient penalty are employed, in which the coefficient for gradient
+penalty is set to be 0.1. Iteratively, 5 updates of the discriminator are performed and
+followed by 1 update of the generators. Except in training PI-GANs, the learning
+rate of Adam optimizer is set to be 10−3 and other hyperparameters of Adam are set
+as default. In training PI-GANs, the learning rate is set to be 10−4, β1 = 0.5 and
+β2 = 0.9 in Adam optimizer for both discriminator and generators.
+
+24
+Z. ZOU AND G. E. KARNIADAKIS
+Training of MH-PINNs, NFs, and PI-GANs was all performed on a single NVIDIA
+TITAN Xp GPU. The L-HYDRA code for TensorFlow implementation along with
+some representative examples will be released on GitHub once the paper is accepted.
+Appendix B. Details for performing Bayesian inference.
+Hamiltonian
+Monte Carlo (HMC) [31] is employed in all Bayesian inference examples for uncer-
+tainty quantification (UQ) while Laplace approximation [18] is only employed in the
+first example. In this paper, HMC with adaptive step size [23] is used, in which the
+initial step size is set to be either 0.1 or 0.01, tuned for better acceptance rate. The
+number of burn-in samples and the number of posterior samples are set to be 1, 000.
+The number of steps for the leapfrog scheme is set to be either 30 or 50, also tuned for
+better acceptance rate. NeuralUQ library [47] for UQ in scientific machine learning is
+used as a tool for physics-informed Bayesian inference in the downstream tasks. The
+ideal acceptance rate in HMC, as discussed in [30, 47], is around 60%. In this paper,
+we found chains with acceptance rate from 50% to 80% acceptable.
+

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+Matching of Everyday Power Supply and Demand with Dynamic Pricing:
+Problem Formalisation and Conceptual Analysis
+Thibaut Théatea,∗, Antonio Suterab, Damien Ernsta,c
+aDepartment of Electrical Engineering and Computer Science, University of Liège, Liège, Belgium
+bHaulogy, Intelligent Systems Solutions, Braine-le-Comte, Belgium
+cInformation Processing and Communications Laboratory, Institut Polytechnique de Paris, Paris, France
+Abstract
+The energy transition is expected to significantly increase the share of renewable energy sources whose production is
+intermittent in the electricity mix. Apart from key benefits, this development has the major drawback of generating a
+mismatch between power supply and demand. The innovative dynamic pricing approach may significantly contribute to
+mitigating that critical problem by taking advantage of the flexibility offered by the demand side. At its core, this idea
+consists in providing the consumer with a price signal which is evolving over time, in order to influence its consumption.
+This novel approach involves a challenging decision-making problem that can be summarised as follows: how to determine
+a price signal maximising the synchronisation between power supply and demand under the constraints of maintaining
+the producer/retailer’s profitability and benefiting the final consumer at the same time? As a contribution, this research
+work presents a detailed formalisation of this particular decision-making problem. Moreover, the paper discusses the
+diverse algorithmic components required to efficiently design a dynamic pricing policy: different forecasting models
+together with an accurate statistical modelling of the demand response to dynamic prices.
+Keywords:
+Matching of supply and demand, dynamic pricing, demand response, power producer/retailer.
+1. Introduction
+Climate change is undeniably a major challenge facing
+humanity in the 21st century [1].
+An ambitious trans-
+formation is required in all sectors to significantly lower
+their respective carbon footprints. Electricity generation
+is no exception, with the burning of fossil fuels, mainly coal
+and gas, being by far the dominant power source in the
+world today [2]. This sector has to undergo an important
+transformation of the global electricity mix by promoting
+power sources with a significantly lower carbon footprint.
+Belonging to that category are nuclear power, hydroelec-
+tricity, biomass or geothermal energy which are relatively
+controllable, but also the energy directly extracted from
+wind and sun which is conversely intermittent in nature.
+Since wind turbines and photovoltaic panels are expected
+to play a key role in the energy transition, solutions are
+required to address their variable production. Interesting
+technical avenues are the interconnection of power grids [3]
+and the development of storage capacities such as battery,
+pumped hydroelectricity or hydrogen [4]. Another promis-
+ing and innovative solution is to influence the behaviour
+of consumers through the use of dynamic pricing (DP), so
+that power supply and demand are better synchronised.
+∗Corresponding author.
+Email addresses: thibaut.theate@uliege.be (Thibaut
+Théate), dernst@uliege.be (Damien Ernst)
+The dynamic pricing approach consists in continuously
+adapting the electricity price that the final consumer has
+to pay in order to influence its consumption behaviour.
+Basically, when demand exceeds supply, the power price
+would be increased in order to take down consumption.
+Conversely, a reduced price would be provided when there
+is excessive production compared to consumption. From a
+graphical perspective, the objective is not only to shift the
+daily consumption curve but also to change its shape in
+order to better overlap with the intermittent production
+curve of renewable energy sources. This is illustrated in
+Figure 1 for a representative situation.
+The innovative dynamic pricing approach relies on two
+important assumptions. Firstly, the final consumer has to
+be equipped with a smart metering device to measure its
+production in real-time and with communication means
+for the price signal. Secondly, the final consumer has to
+be able to provide a certain amount of flexibility regarding
+its power consumption. Moreover, it has to be sufficiently
+receptive to the incentives offered to reduce its electricity
+bill in exchange for a behaviour change. If these require-
+ments are met, the major strength of the dynamic pricing
+approach is its potential benefits for both the consumer
+and the producer/retailer. Moreover, these benefits would
+not only be in terms of economy, but also potentially in
+terms of ecology and autonomy. In fact, dynamic prices
+reward the flexibility of the demand side.
+arXiv:2301.11587v1  [q-fin.TR]  27 Jan 2023
+
+12:00
+06:00
+18:00
+12:00
+06:00
+18:00
+Supply
+Demand
+Time
+Time
+Power
+Power
+Figure 1: Illustration of the dynamic pricing approach’s potential
+to shift and change the shape of a typical daily consumption curve
+(blue) so that there is a better synchronisation with the daily inter-
+mittent production curve of renewable energy sources (red).
+The contributions of this research work are twofold.
+Firstly, the complex decision-making problem faced by
+a producer/retailer willing to develop a dynamic pricing
+strategy is presented and rigorously formalised. Secondly,
+the diverse algorithmic components required to efficiently
+design a dynamic pricing policy are thoroughly discussed.
+To the authors’ knowledge, demand response via dynamic
+pricing has received considerable attention from the re-
+search community, but from the perspective of the demand
+side alone. Therefore, the present research may be consid-
+ered as a pioneer work studying dynamic pricing from the
+perspective of the supply side for taking advantage of the
+flexibility of the power consumers.
+This research paper is structured as follows. First of
+all, the scientific literature about both dynamic pricing
+and demand response is concisely reviewed in Section 2.
+Then, Section 3 presents a detailed formalisation of the
+decision-making problem behind the novel dynamic pric-
+ing approach. Afterwards, Section 4 analyses the algorith-
+mic components necessary for the development of dynamic
+pricing policies. Subsequently, a fair performance assess-
+ment methodology is introduced in Section 5 to quanti-
+tatively evaluate the performance of a dynamic pricing
+policy. To end this paper, Section 6 discusses interesting
+research avenues for future work and draws conclusions.
+2. Literature review
+Over the last decade, the management of the demand
+side in the scope of the energy transition has received in-
+creasing attention from the research community. In fact,
+there exist multiple generic approaches when it comes to
+demand response. Without getting into too many details,
+the scientific literature includes some surveys summarising
+and discussing the different techniques available together
+with their associated challenges and benefits [5, 6, 7, 8, 9].
+In this research work, the focus is exclusively set on the
+demand response induced by dynamic power prices.
+As previously mentioned, the scientific literature about
+demand response via dynamic pricing is primarily focused
+on the perspective of the demand side. Multiple techniques
+have already been proposed to help the consumer provide
+flexibility and take advantage of behavioural changes to
+lower its electricity bill. For instance, [10] presents a power
+scheduling method based on a genetic algorithm to opti-
+mise residential demand response via an energy manage-
+ment system, so that the electricity cost is reduced. In [11],
+a technique based on dynamic programming is introduced
+for determining the optimal schedule of residential con-
+trollable appliances in the context of time-varying power
+pricing. One can also mention [12] that proposes an energy
+sharing model with price-based demand response for mi-
+crogrids of peer-to-peer prosumers. The approach is based
+on a distributed iterative algorithm and has been shown
+to lower the prosumers’ costs and improve the sharing of
+photovoltaic energy. More recently, (deep) reinforcement
+learning techniques have been proven to be particularly
+relevant for controlling the residential demand response in
+the context of dynamic power prices [13, 14].
+On the contrary, the question of inducing a residential
+demand response based on a dynamic pricing approach
+from the perspective of the supply side has not received
+a lot of attention from the research community yet. Still,
+there are a few works in the scientific literature about the
+mathematical modelling of the demand response caused by
+dynamic power prices, which is a key element in achieving
+that objective. To begin with, [15] presents a simulation
+model highlighting the evolution of electricity consump-
+tion profiles when shifting from a fixed tariff to dynamic
+power prices. The same objective is pursued by [16] which
+introduces a fully data-driven approach relying on the data
+collected by smart meters and exogenous variables. The
+resulting simulation model is based on consumption pro-
+files clustering and conditional variational autoencoders.
+Alternatively, [17] presents a functional model of residen-
+tial power consumption elasticity under dynamic pricing to
+assess the impact of different electricity price levels, based
+on a Bayesian probabilistic approach. In addition to these
+mathematical models, one can also mention some real-life
+experiments conducted to assess the responsiveness of res-
+idential electricity demand to dynamic pricing [18, 19].
+2
+
+3. Problem formalisation
+This section presents a mathematical formalisation of
+the challenging sequential decision-making problem related
+to the dynamic pricing approach for inducing a residential
+demand response.
+To begin with, the contextualisation
+considered for studying this particular problem is briefly
+described, followed by an overview of the decision-making
+process. Then, a discretisation of the continuous timeline
+is introduced. Subsequently, the formal definition of a dy-
+namic pricing policy is presented. Lastly, the input and
+output spaces of a dynamic pricing policy are described,
+together with the objective criterion.
+3.1. Contextualisation
+As previously hinted, this research work focuses on the
+interesting real-case scenario of a producer/retailer whose
+production portfolio is composed of an important share of
+renewable energy sources such as wind turbines and photo-
+voltaic panels. Because of the substantial intermittency of
+these generation assets, a strong connection to the energy
+markets is required in order to fully satisfy its customers
+regardless of the weather. Nevertheless, the consumers are
+assumed to be well informed and willing to adapt their be-
+haviour in order to consume renewable energy rather than
+electricity purchased on the market whose origin may be
+unknown. Within this particular context, the benefits of
+the dynamic pricing approach taking advantage of the con-
+sumers’ flexibility are maximised. Indeed, the insignificant
+marginal cost associated with these intermittent renew-
+able energy sources coupled with their low carbon foot-
+print make this innovative approach interesting from an
+economical perspective for both supply and demand sides,
+but also in terms of ecology. Moreover, the autonomy of
+the producer/retailer is expected to be reinforced by low-
+ering its dependence on the energy markets. At the same
+time, dependence on fossil fuels may be reduced as well.
+In this research work, the predicted difference between
+power production and consumption is assumed to be fully
+secured in the day-ahead electricity market. Also called
+spot market, the day-ahead market has an hourly resolu-
+tion and is operated once a day for all hours of the follow-
+ing day via a single-blind auction. In other words, trading
+power for hour H of day D has to be performed ahead on
+day D − 1 between 00:00 AM (market opening) and 12:00
+AM (market closure).
+Therefore, the energy is at best
+purchased 12 hours (00:00 AM of day D) up to 35 hours
+(11:00 PM of day D) before the actual delivery of power.
+Apart from the day-ahead electricity market, it is assumed
+that there are no trading activities on the future/forward
+nor intraday markets. Nevertheless, if there remains an
+eventual mismatch between production and consumption
+at the time of power delivery, the producer/retailer would
+be exposed to the imbalance market. In this case, the so-
+called imbalance price has to be inevitably paid as com-
+pensation for pushing the power grid off balance.
+3.2. Decision-making process overview
+The decision-making problem studied in this research
+work is characterised by a particularity: a variable time lag
+between the moment a decision is made and the moment it
+becomes effective. As previously explained, any remaining
+difference between production and consumption after de-
+mand response has to ideally be traded on the day-ahead
+market. The purpose of this assumption is to limit the ex-
+posure of the producer/retailer to the imbalance market.
+For this reason, the price signal sent to the consumer on
+day D has to be generated before the closing of the day-
+ahead market on day D − 1. Additionally, it is assumed
+that the price signal cannot be refreshed afterwards.
+Basically, the decision-making problem at hand can be
+formalised as follows. The core objective is to determine
+a decision-making policy, denoted Π, mapping at time τ
+input information of diverse nature Iτ to the electricity
+price signal Sτ to be sent to the consumers over a future
+time horizon well-defined:
+Sτ = Π(Iτ),
+(1)
+where:
+• Iτ represents the information vector gathering all the
+available information (of diverse nature) at time τ
+which may be helpful to make a relevant dynamic
+pricing decision,
+• Sτ represents a set of electricity prices generated at
+time τ and shaping the dynamic price signal over a
+well-defined future time horizon.
+The dynamic pricing approach from the perspective of
+the supply side belongs to a particular class of decision-
+making problems:
+automated planning and scheduling.
+Contrarily to conventional decision-making outputting one
+action at a time, planning decision-making is concerned
+with the generation of a sequence of actions.
+In other
+words, a planning decision-making problem requires to
+synthesise in advance a strategy or plan of actions over
+a certain time horizon. Formally, the decision-making has
+to be performed at a specific time τ about a control vari-
+able over a future time horizon beginning at time τi > τ
+and ending at time τf > τi. In this case, the decision-
+making is assumed to be performed just before the closing
+of the day-ahead market at 12:00 AM to determine the
+price signal to be sent to the consumers throughout the
+entire following day (from 00:00 AM to 11:59 PM).
+In the next sections, a more accurate and thorough
+mathematical formalisation of the dynamic pricing prob-
+lem from the perspective of the supply side is presented.
+Moreover, the planning problem previously introduced is
+cast into a sequential decision-making problem. Indeed,
+this research paper intends to focus on a decision-making
+policy outputting a single price from the signal Sτ at a
+time based on a subset of the information vector Iτ.
+3
+
+00:00
+12:00
+𝑦� 𝑦�
+𝑦��
+Closing of the
+day-ahead market
+𝑥�
+𝑥�
+𝑥�
+𝑥��
+Dynamic pricing
+policy π
+𝑦�
+Forecasts
+𝑝�
+�
+𝑐�
+�
+𝜆�
+�
+𝑐�
+�
+Time
+Power
+Time
+Time
+Time
+Power
+Power
+Price
+Demand 
+reponse 
+model
+Figure 2: Illustration of the formalised decision-making problem related to dynamic pricing from the perspective of the supply side. The
+notations xt and yt represent the inputs and outputs of a dynamic pricing policy π, which are not shown concurrent on the timeline since
+the decision-making occurs multiple hours before the application of the dynamic pricing signal. The time axis of the four plots represents the
+complete following day for which the dynamic prices are generated. The mathematical notations pF
+t , cF
+t and λF
+t respectively represent the
+forecast production, consumption and day-ahead market price for the time step t. Finally, the quantity c′
+t is the predicted consumption at
+time step t after taking into consideration the dynamic pricing signal.
+3.3. Timeline discretisation
+Theoretically, the dynamic electricity price signal sent
+to the consumer could be continuously changing over time.
+More realistically, this research work adopts a discretisa-
+tion of the continuous timeline so that this power price
+is adapted at regular intervals.
+Formally, this timeline
+is discretised into a number of time steps t spaced by a
+constant duration ∆t. If the duration ∆t is too large, the
+synchronisation improvement between supply and demand
+will probably be of poor quality. Conversely, lowering the
+value of the duration ∆t increases the complexity of the
+decision-making process, and a too high update frequency
+may even confuse the consumer. There is a trade-off to
+be found concerning this important parameter. In this re-
+search work, the dynamic price signal is assumed to change
+once per hour, meaning that ∆t is equal to one hour. This
+choice is motivated by the hourly resolution of the day-
+ahead market, which has proven to be an appropriate com-
+promise over the years for matching power production and
+consumption. Another relevant discretisation choice could
+be to have a price signal which is updated every quarter of
+an hour. In the rest of this research paper, the increment
+(decrement) operations t + 1 (t − 1) are used to model the
+discrete transition from time step t to time step t + ∆t
+(t − ∆t), for the sake of clarity.
+3.4. Dynamic pricing policy
+Within the context previously described, a dynamic
+pricing planning policy Π consists of the set of rules used
+to make a decision regarding the future price signal sent to
+the consumers over the next day. This planning policy can
+be decomposed into a set of 24 dynamic pricing decision-
+making policies π outputting a single electricity price for
+one hour of the following day.
+Mathematically, such a
+dynamic pricing strategy can be defined as a programmed
+policy π : X → Y, either deterministic or stochastic, which
+outputs a decision yt ∈ Y for time step t based on some
+input information xt ∈ X so as to maximise an objective
+criterion. The input xt is derived from the information vec-
+tor Iτ associated with the decision-making for time step t,
+after potential preprocessing operations. The price signal
+Sτ is composed of 24 dynamic pricing policy outputs yt.
+In the rest of this research work, the time at which
+the decision-making does occur should not be confused
+with the time at which the dynamic price signal is active
+(charging for energy consumption). The proposed formal-
+isation assumes that the time step t refers to the time at
+which the dynamic price is active, not decided. Therefore,
+the decision-making of the dynamic pricing policy for time
+step t (yt = π(xt)) is in fact performed hours in advance
+of time step t. This complexity is illustrated in Figure 2
+describing the formalised decision-making problem.
+4
+
+I.3.5. Input of a dynamic pricing policy
+The input space X of a dynamic pricing policy π com-
+prises all the available information which may help to make
+a relevant decision about future electricity prices so that
+an appropriate demand response is induced.
+Since the
+decision-making occurs 12 up to 35 hours in advance of the
+price signal delivery, this information mainly consists of
+forecasts and estimations that are subject to uncertainty.
+As depicted in Figure 2, the dynamic pricing policy input
+xt ∈ X refers to the decision-making occurring at time
+τ = t − h with h ∈ [12, 35] about the dynamic pricing
+signal delivered to the consumer at time step t. In fact,
+the quantity Iτ may be seen as the information contained
+in the 24 inputs xt for t ∈ {τ + 12, ..., τ + 35}. Formally,
+the input xt ∈ X is decided to be defined as follows:
+xt = {P F
+t , CF
+t , ΛF
+t , Yt, M},
+(2)
+where:
+• P F
+t
+= {pF
+t+ϵ ∈ R+ | ϵ = −k, ..., k} represents a set
+of forecasts for the power production within a time
+window centred around time step t and of size k,
+• CF
+t = {cF
+t+ϵ ∈ R+ | ϵ = −k, ..., k} represents a set of
+forecasts for the power consumption within a time
+window centred around time step t and of size k,
+• ΛF
+t = {λF
+t+ϵ ∈ R | ϵ = −k, ..., k} represents a set of
+forecasts for the day-ahead market prices within a
+window centred around time step t and of size k,
+• Yt = {yt−ϵ ∈ R | ϵ = 1, ..., k} represents the series of
+k previous values for the dynamic price signal sent
+to the final consumer,
+• M is a mathematical model of the demand response
+to be expected from the consumption portfolio, with
+the required input information.
+The different forecasting models and the challenging
+modelling of the consumption portfolio demand response
+are discussed in more details in Section 4.
+3.6. Output of a dynamic pricing policy
+The output space Y of a dynamic pricing policy π only
+includes the future price signal to be sent to the consumer.
+Formally, the dynamic pricing policy output yt ∈ Y, which
+represents the electricity price to be paid by the consumer
+for its power consumption at time step t, is mathematically
+defined as follows:
+yt = et,
+(3)
+where et ∈ R represents the dynamic electricity price to
+be paid by the demand side for its power consumption
+at time step t.
+Out of the scope of this research work
+is the presentation of this price signal so that the impact
+on the final consumer is maximised. Indeed, the way of
+communicating the output of the dynamic pricing policy
+has to be adapted to the audience, be it humans with
+different levels of electricity market expertise or algorithms
+(energy management systems).
+3.7. Objective criterion
+The dynamic pricing approach can provide multiple
+benefits, in terms of economy, ecology but also autonomy.
+Consequently, the objective criterion to be maximised by
+a dynamic pricing policy π is not trivially determined. In
+fact, several core objectives can be clearly identified:
+• maximising the match between supply and demand,
+• minimising the carbon footprint of power generation,
+• minimising the electricity costs for the consumer,
+• maximising the revenue of the producer/retailer.
+Although some objectives overlap, these four criteria
+are not completely compatible. For instance, maximising
+the synchronisation between power supply and demand is
+equivalent to minimising the carbon footprint associated
+with the generation of electricity. Indeed, the production
+portfolio of the producer/retailer being mainly composed
+of intermittent renewable energy sources, its energy has a
+reduced carbon footprint compared to the electricity that
+can be purchased on the day-ahead market whose origin is
+unknown. On the contrary, maximising the revenue of the
+producer/retailer will obviously not lead to a minimised
+electricity bill for the consumer. This research work makes
+the choice to prioritise the maximisation of the synchroni-
+sation between supply and demand, and equivalently the
+minimisation of the carbon footprint, while translating the
+other two core objectives into relevant constraints. Firstly,
+the costs for the consumer have to be reduced with respect
+to the situation without dynamic pricing. Secondly, the
+profitability of the producer/retailer has to be guaranteed.
+Formally, the objective criterion to be optimised by a
+dynamic pricing policy π can be mathematically defined
+as the following. First of all, the main target to evaluate
+is the synchronisation between supply and demand, which
+can be quantitatively assessed through the deviation ∆T .
+This quantity has to ideally be minimised, and can be
+mathematically expressed as follows:
+∆T =
+T −1
+�
+t=0
+|pt − ct|,
+(4)
+where:
+• t = 0 corresponds to the first electricity delivery hour
+of a new day (00:00 AM),
+• T is the time horizon considered, which should be a
+multiple of 24 to have full days,
+• pt is the actual power production (not predicted)
+from the supply side at time step t,
+• ct is the actual power consumption (not predicted)
+from the demand side at time step t.
+5
+
+Afterwards, the first constraint concerning the reduced
+costs for the consumer has to be modelled mathematically.
+This is achieved via the electricity bill BT paid by the
+consumer over the time horizon T, which can be expressed
+as the following:
+BT =
+T −1
+�
+t=0
+ct yt .
+(5)
+As previously explained, the consumer power bill BT
+should not exceed that obtained without dynamic pricing.
+In that case, the consumer is assumed to pay a price et,
+which can for instance be a fixed tariff or a price indexed
+on the day-ahead market price.
+The situation without
+dynamic pricing is discussed in more details in Section 5.
+Consequently, the first constraint can be mathematically
+expressed as follows:
+T −1
+�
+t=0
+ct yt ≤
+T −1
+�
+t=0
+ct et ,
+(6)
+where ct is the power consumption from the demand side
+at time step t without dynamic pricing.
+Then, the second constraint is about the profitability
+of the producer/retailer, which is achieved if its revenue
+exceeds its costs. The revenue RT of the producer/retailer
+over the time horizon T can be mathematically expressed
+as the following:
+RT =
+T −1
+�
+t=0
+�
+ct yt − (c′
+t − pF
+t ) λt − (ct − pt) it
+�
+,
+(7)
+where:
+• λt is the actual power price (not predicted) on the
+day-ahead market at time step t,
+• it is the actual imbalance price (not predicted) on
+the imbalance market at time step t,
+• c′
+t is the predicted power consumption at time step t
+after demand response to the dynamic prices, based
+on the demand response mathematical model M.
+The first term corresponds to the payment of the cus-
+tomers for their electricity consumption. The second term
+is the revenue or cost induced by the predicted mismatch
+between supply and demand, which is traded on the day-
+ahead market. The last term is the cost or revenue caused
+by the remaining imbalance between supply and demand,
+which has to be compensated in the imbalance market.
+The total costs incurred by the producer/retailer at
+each time step t can be decomposed into both fixed costs
+FC and marginal costs MC. In this particular case, the
+marginal costs of production are assumed to be negligible
+since the production portfolio is composed of intermittent
+renewable energy sources such as wind turbines and pho-
+tovoltaic panels. Therefore, the second constraint can be
+mathematically expressed as follows:
+T −1
+�
+t=0
+�
+ct yt − (c′
+t − pF
+t ) λt − (ct − pt) it
+�
+≥ FC T .
+(8)
+Finally, the complete objective criterion to be opti-
+mised by a dynamic pricing policy can be mathematically
+expressed as follows:
+minimise
+π
+T −1
+�
+t=0
+|pt − ct|,
+subject to
+RT ≥ FC T ,
+BT ≤
+T −1
+�
+t=0
+ct et .
+(9)
+4. Algorithmic components discussion
+This section presents a thorough discussion about the
+different algorithmic modules required to efficiently design
+a dynamic pricing policy from the perspective of the supply
+side. Firstly, the different forecasting blocks are rigorously
+analysed. Secondly, the modelling of the demand response
+induced by dynamic prices is discussed. Lastly, the proper
+management of uncertainty is considered.
+In parallel, for the sake of clarity, Figure 3 highlights
+the interconnections between the different algorithmic com-
+ponents in the scope of a dynamic pricing policy from the
+perspective of the supply side.
+Moreover, Algorithm 1
+provides a thorough description of the complete decision-
+making process for the dynamic pricing problem at hand.
+The complexity of the variable time lag between decision-
+making and application is highlighted. Assuming that the
+decision-making occurs once a day at 12:00 AM just be-
+fore the closing of the day-ahead market for all hours of
+the following day, the dynamic price at time step t is de-
+cided hours in advance at time step t − [12 + (t%24)] with
+the symbol % representing the modulo operation.
+4.1. Production forecasting
+The first forecasting block to be discussed concerns the
+production of intermittent renewable energy sources such
+as wind turbines and photovoltaic panels. Indeed, having
+access to accurate predictions about the future output of
+the production portfolio is key to the performance of a
+dynamic pricing policy from the perspective of the supply
+side. As previously explained in Section 3.4, the forecasts
+have to be available one day ahead before the closing of
+the day-ahead electricity market for all hours of the fol-
+lowing day. Naturally, the generation of such predictions
+introduces uncertainty, a complexity that has to be taken
+into account to design sound dynamic pricing policies.
+6
+
+𝑝�
+�
+𝑐�
+�
+𝜆�
+�
+Time
+Time
+Time
+Power
+Power
+Price
+𝑐�
+�
+Time
+Power
+𝑐�
+�
+Time
+Power
+𝑦�
+Time
+Price
+Figure 3: Illustration of the complete decision-making process related to dynamic pricing from the perspective of the supply side, with the
+connections between the different algorithmic components highlighted.
+Algorithm 1 Dynamic pricing complete decision-making process
+The decision-making occurs once per day before the closing of the day-ahead market at 12:00 AM for all hours of the next day.
+The decision-making for the dynamic price of time step t occurs at time step t − [12 + (t%24)].
+for τ = −12 to T − 12 do
+Check whether the time is 12:00 AM to proceed to the decision-making.
+if (τ + 12)%24 = 0 then
+for t = τ + 12 to τ + 36 do
+Gather the available information for production forecasting xP
+t = {W F
+t , AF
+t , IP
+t }.
+Gather the available information for consumption forecasting xC
+t = {W F
+t , Tt, IC
+t }.
+Gather the available information for day-ahead market price forecasting xM
+t
+= {xP
+t , xC
+t , GF
+t , Mt, IM
+t }.
+Forecast production at time step t: pF
+t = FP
+�
+xP
+t
+�
+.
+Forecast consumption at time step t: cF
+t = FC
+�
+xC
+t
+�
+.
+Forecast the day-ahead market price at time step t: λF
+t = FM
+�
+xM
+t
+�
+.
+end for
+for t = τ + 12 to τ + 36 do
+Gather the input information for the dynamic pricing policy xt = {P F
+t , CF
+t , ΛF
+t , Yt, M}.
+Make a dynamic pricing decision for time step t: yt = π(xt).
+end for
+Announce the dynamic prices for all hours of the following day {yt | t = τ + 12, ..., τ + 35}.
+end if
+end for
+7
+
+Dynamic pricing policy TX
+MProduction forecasting FjConsumption forecasting F'Market price forecasting FDemand
+resnonse
+ model MFormally, the forecasting model associated with the
+output of the production portfolio is denoted FP . Its input
+space XP comprises every piece of information that may
+potentially have an impact on the generation of electricity
+from intermittent renewable energy sources such as wind
+turbines and photovoltaic panels for a certain time period.
+Its output space YP is composed of a forecast regarding the
+power generation from the production portfolio for that
+same time period. Mathematically, the forecasting model
+input xP
+t ∈ XP and output yP
+t ∈ YP at time step t can be
+expressed as follows:
+xP
+t = {W F
+t , AF
+t , IP
+t },
+(10)
+yP
+t = pF
+t ,
+(11)
+where:
+• W F
+t
+represents various weather forecasts related to
+the power production of intermittent renewable en-
+ergy sources such as wind turbines and photovoltaic
+panels (wind speed/direction, solar irradiance, etc.)
+at the time step t,
+• AF
+t represents predictions about the available capac-
+ity of the production portfolio at time step t, which
+may be impacted by scheduled maintenance, repairs,
+or other similar constraints,
+• IP
+t
+represents any additional information that may
+help to accurately forecast the future power gener-
+ation of the producer/retailer’s production portfolio
+at time step t.
+In the scientific literature, the current state of the art
+for forecasting the power production of intermittent renew-
+able energy sources is mainly based on deep learning tech-
+niques together with some data cleansing processes and
+data augmentation approaches.
+The best architectures
+are recurrent neural networks (RNN), convolutional neural
+networks (CNN) and transformers [20, 21, 22, 23, 24].
+4.2. Consumption forecasting
+The objective of the next important forecasting model
+deserving a discussion is to accurately predict the future
+power demand of the consumption portfolio before any
+demand response phenomenon is induced. Since the main
+goal of a dynamic pricing policy is to maximise the syn-
+chronisation between supply and demand, electricity load
+forecasts are of equal importance to electricity generation
+predictions. Similarly to the latter, the portfolio consump-
+tion forecasts are assumed to be generated one day ahead
+just before the closing of the day-ahead market for all 24
+hours of the following day. Additionally, the uncertainty
+associated with these predictions has to be seriously taken
+into account for the success of the dynamic pricing policy.
+From a more formal perspective, the forecasting model
+responsible for predicting the future electricity load of the
+consumption portfolio is denoted FC. Its input space XC
+includes all the information that may have an influence on
+the residential electricity consumption for a certain time
+period. Its output space YC comprises a forecast of the
+power used by the consumption portfolio for that same
+time period. Mathematically, the consumption forecasting
+model input xC
+t ∈ XC and output yC
+t ∈ YC at time step t
+can be expressed as the following:
+xC
+t = {W F
+t , Tt, IC
+t },
+(12)
+yC
+t = cF
+t ,
+(13)
+where:
+• W F
+t
+represents various weather forecasts related to
+the residential electricity consumption (temperature,
+hygrometry, etc.) at the time step t,
+• Tt represents diverse characteristics related to the
+time step t (hour, weekend, holiday, season, etc.),
+• IC
+t represents supplementary information that could
+potentially have an influence on the residential power
+consumption at time step t.
+Similarly to renewable energy production forecasting,
+the state-of-the-art approaches for predicting the residen-
+tial electricity load in the short term are mostly related
+to deep learning techniques with preprocessed augmented
+data: RNN, CNN, and transformers [25, 26, 27, 28, 22].
+4.3. Market price forecasting
+The last forecasting block to be discussed concerns
+the future day-ahead electricity market prices. Contrarily
+to the forecasting of power production and consumption,
+these price predictions are not critical to the success of a
+dynamic pricing policy from the perspective of the supply
+side. Still, having access to quality forecasts for the fu-
+ture day-ahead market prices remains important in order
+to satisfy the constraints related to the profitability of the
+producer/retailer as well as the reduced electricity costs for
+the consumer. Once again, the predictions are assumed to
+be made just before the closing of the day-ahead market.
+Moreover, the uncertainty associated with these forecasts
+has to be taken into consideration.
+Formally, the forecasting model related to the future
+day-ahead electricity market prices is denoted FM.
+Its
+input space XM includes every single piece of information
+which may potentially explain the future electricity price
+on the day-ahead market for a certain hour. Its output
+space YM comprises a forecast of the day-ahead market
+price for that same hour. Mathematically, both forecasting
+8
+
+model input xM
+t
+∈ XM and output yM
+t
+∈ YM at time step
+t can be expressed as follows:
+xM
+t
+= {xP
+t , xC
+t , GF
+t , Mt, IM
+t },
+(14)
+yM
+t
+= λF
+t ,
+(15)
+where:
+• GF
+t represents forecasts about the state of the power
+grid as a whole (available production capacity, trans-
+mission lines, etc.) at the time step t,
+• Mt represents diverse information in various markets
+related to energy (power, carbon, oil, gas, coal, etc.)
+in neighbouring geographical areas at time step t,
+• IM
+t
+represents any extra piece of information that
+may help to predict the future electricity price on
+the day-ahead market at time step t.
+Once again, the scientific literature reveals that the
+state-of-the-art approaches for day-ahead power market
+price forecasting are mostly based on innovative machine
+learning techniques [29, 30, 31, 32, 33].
+4.4. Demand response modelling
+Another essential algorithmic component is the math-
+ematical modelling of the residential demand response to
+dynamic prices. In order to make relevant dynamic pric-
+ing decisions, an estimation of the impact of the electricity
+price on the consumer’s behaviour is necessary. In fact,
+two important characteristics have to be studied:
+The residential power consumption elasticity. This
+quantity measures the average percentage change of the
+residential power consumption in response to a percentage
+change in the electricity price. In other words, the elastic-
+ity captures the willingness of the consumer to adapt its
+behaviour when the price of electricity either increases or
+decreases. This elasticity is critical to the dynamic pricing
+approach, since it assesses the receptiveness of the con-
+sumers to dynamic prices. In fact, the residential power
+consumption elasticity can be considered as a quantitative
+indicator of the potential of the dynamic pricing approach.
+The electricity load temporal dependence.
+Time
+plays an important role in power consumption.
+Firstly,
+the consumer’s behaviour is highly dependent on the time
+of the day. The tendency to adapt this behaviour is also
+expected to be time-dependent. Therefore, the residential
+power consumption elasticity has to be a function of the
+time within a day, among other things. Secondly, a higher
+electricity price does not simply reduce the demand as with
+other commodities, but rather shifts part of the consump-
+tion earlier and/or later in time. This phenomenon reflects
+a complex temporal dependence for power consumption,
+which has to be accurately modelled in order to design a
+performing dynamic pricing policy.
+Formally, the mathematical model of the residential
+demand response is denoted M.
+Its input space XD is
+composed of the predicted power consumption before any
+demand response and the dynamic prices to be sent to the
+consumers for several hours before and after the time pe-
+riod analysed, together with information about that time
+period. Its output space YD comprises the predicted power
+consumption after demand response to dynamic prices for
+that same time period. Mathematically, both demand re-
+sponse model input xD
+t ∈ XD and output yD
+t ∈ YD at time
+step t can be expressed as the following:
+xD
+t = {CF
+t , Y ′
+t , Tt},
+(16)
+yD
+t = c′
+t,
+(17)
+where Y ′
+t = {yt+ϵ ∈ R | ϵ = −k, ..., k} is the dynamic price
+signal within a time window centred around time step t
+and of size k from which the demand response is induced.
+As far as the scientific literature about the modelling of
+demand response to dynamic prices is concerned, this in-
+teresting topic has not yet received a lot of attention from
+the research community. Still, there exists a few sound
+works presenting demand response models and assessing
+the receptiveness of the consumers to dynamic power prices
+[15, 16, 17, 18, 19], as explained in Section 2.
+4.5. Uncertainty discussion
+As previously hinted, a dynamic pricing policy has to
+make its decisions based on imperfect information. Indeed,
+multiple forecasts for the electricity price, production and
+consumption have to be generated 12 up to 35 hours in
+advance. Naturally, these predictions comes with a level
+of uncertainty that should not be neglected.
+Moreover,
+accurately modelling the residential demand response to
+dynamic prices is a particularly challenging task. Because
+of both the random human nature and the difficulty to
+fully capture the consumers’ behaviour within a mathe-
+matical model, a notable level of uncertainty should also
+be considered at this stage. Therefore, multiple sources of
+uncertainty can be identified in the scope of the dynamic
+pricing decision-making problem at hand, and a proper
+management of this uncertainty is necessary.
+A stochastic reasoning is recommended to make sound
+dynamic pricing decisions despite this substantial level of
+uncertainty. Instead of considering each uncertain variable
+(production, consumption, price, demand response) with
+a probability of 1, the full probability distribution behind
+these quantities has to be estimated and exploited. Based
+on this information, the risk associated with uncertainty
+may be mitigated. Moreover, safety margins may also con-
+tribute to reduce this risk, but potentially at the expense
+of a lowered performance. In fact, there generally exists
+9
+
+a trade-off between performance and risk, in line with the
+adage: with great risk comes great reward.
+5. Performance assessment methodology
+This section presents a methodology for quantitatively
+assessing the performance of a dynamic pricing policy in
+a comprehensive manner.
+As explained in Section 3.7,
+several disjoint objectives can be clearly identified.
+For
+the sake of completeness, this research work presents three
+quantitative indicators, one for each objective. The rela-
+tive importance of these indicators is left to the discretion
+of the reader according to its main intention among the
+different objectives previously defined.
+The performance indicators proposed are based on the
+comparison with the original situation without dynamic
+pricing. In this case, the consumer is assumed to be fully
+ignorant about the mismatch problem between supply and
+demand. No information is provided to the customers of
+the producer/retailer, which consequently have an unin-
+fluenced consumption behaviour. The price of electricity
+et is freely determined by the producer/retailer. It may
+for instance be a fixed tariff, or a price indexed on the
+day-ahead market price:
+et = α λt + β ,
+(18)
+where α and β are parameters to be set by the retailer.
+Firstly, the impact of a dynamic pricing policy on the
+synchronisation between power supply and demand can be
+assessed through the performance indicator S quantifying
+the relative evolution of the deviation ∆T . This quantity
+is mathematically expressed as follows:
+S = 100 ∆T − ∆T
+∆T
+,
+(19)
+∆T =
+T −1
+�
+t=0
+|pt − ct| ,
+(20)
+where ∆T represents the lack of synchronisation between
+supply and demand without dynamic pricing. Therefore,
+the quantity S has ideally to be maximised, with a perfect
+synchronisation between supply and demand leading to a
+value of 100% reduction in deviation.
+Secondly, the consequence for the consumer regarding
+its electricity bill can be evaluated with the quantity B
+which informs about the relative evolution of this power
+bill. It can be mathematically computed as the following:
+B = 100 BT − BT
+BT
+,
+(21)
+where BT = �T −1
+t=0 ct et represents the electricity bill paid
+by the consumer without dynamic pricing. Since the per-
+formance indicator B represents the percentage reduction
+in costs, it has to ideally be maximised.
+Lastly, the enhancement in terms of revenue for the
+producer/retailer can be efficiently quantified thanks to
+the performance indicator R. This quantity represents the
+relative evolution of the producer/retailer revenue and can
+be mathematically expressed as follows:
+R = 100 RT − RT
+RT
+,
+(22)
+RT =
+T −1
+�
+t=0
+�
+ct et − (cF
+t − pF
+t ) λt − (ct − pt) it
+�
+,
+(23)
+where RT represents the producer/retailer revenue with-
+out dynamic pricing. Obviously, the performance indicator
+R has to ideally be maximised.
+6. Conclusion
+This research paper presents a detailed formalisation of
+the decision-making problem faced by a producer/retailer
+willing to adopt a dynamic pricing approach, in order to
+induce an appropriate residential demand response. Three
+core challenges are highlighted by this formalisation work.
+Firstly, the objective criterion maximised by a dynamic
+pricing policy is not trivially defined, since different goals
+that are not compatible can be clearly identified. Secondly,
+several complex algorithmic components are necessary for
+the development of a performing dynamic pricing policy.
+One can for instance mention different forecasting blocks,
+but also a mathematical model of the residential demand
+response to dynamic prices. Thirdly, the dynamic pricing
+decisions have to be made based on imperfect information,
+because this particular decision-making problem is highly
+conditioned by the actual uncertainty for the future.
+Several avenues are proposed for future work. In fact,
+the natural extension of the present research is to design
+innovative dynamic pricing policies from the perspective
+of the supply side based on the formalisation performed.
+While the present research paper exclusively focuses on
+the philosophy and conceptual analysis of the approach,
+there remain practical concerns that need to be properly
+addressed in order to achieve performing decision-making
+policies. To achieve that, a deeper analysis of the scientific
+literature about each algorithmic component discussed in
+Section 4 is firstly welcomed, in order to identify and re-
+produce the state-of-the-art techniques within the context
+of interest. Then, different approaches have to be investi-
+gated for the design of the dynamic pricing policy itself.
+One can for instance mention, among others, the stochastic
+optimisation and deep reinforcement learning techniques.
+Finally, the dynamic pricing policies developed have to be
+rigorously evaluated, analysed, and compared by taking
+advantage of real-life experiments.
+10
+
+Acknowledgements
+Thibaut Théate is a Research Fellow of the F.R.S.-
+FNRS, of which he acknowledges the financial support.
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+DiffTalk: Crafting Diffusion Models for Generalized Talking Head Synthesis
+Shuai Shen1
+Wenliang Zhao1
+Zibin Meng1
+Wanhua Li1
+Zheng Zhu2
+Jie Zhou1
+Jiwen Lu1
+1Tsinghua University
+2PhiGent Robotics
+…
+Figure 1. We present a crafted conditional Diffusion model for generalized Talking head synthesis (DiffTalk). Given a driven audio, the
+DiffTalk is capable of synthesizing high-fidelity and synchronized talking videos for multiple identities without further fine-tuning.
+Abstract
+Talking head synthesis is a promising approach for the
+video production industry.
+Recently, a lot of effort has
+been devoted in this research area to improve the gener-
+ation quality or enhance the model generalization. How-
+ever, there are few works able to address both issues simul-
+taneously, which is essential for practical applications. To
+this end, in this paper, we turn attention to the emerging
+powerful Latent Diffusion Models, and model the Talking
+head generation as an audio-driven temporally coherent
+denoising process (DiffTalk).
+More specifically, instead
+of employing audio signals as the single driving factor,
+we investigate the control mechanism of the talking face,
+and incorporate reference face images and landmarks as
+conditions for personality-aware generalized synthesis. In
+this way, the proposed DiffTalk is capable of producing
+high-quality talking head videos in synchronization with the
+source audio, and more importantly, it can be naturally gen-
+eralized across different identities without any further fine-
+tuning.
+Additionally, our DiffTalk can be gracefully tai-
+lored for higher-resolution synthesis with negligible extra
+computational cost. Extensive experiments show that the
+proposed DiffTalk efficiently synthesizes high-fidelity audio-
+driven talking head videos for generalized novel identi-
+ties. For more video results, please refer to this demon-
+stration https://cloud.tsinghua.edu.cn/f/
+e13f5aad2f4c4f898ae7/.
+1. Introduction
+Talking head synthesis is a challenging and promising re-
+search topic, which aims to synthesize a talking video with
+given audio. This technique is widely applied in various
+practical scenarios including animation, virtual avatars, on-
+line education, and video conferencing [4,44,47,50,52].
+Recently a lot of effort has been devoted to this re-
+search area to improve the generation quality or enhance
+the model generalization.
+Among these existing main-
+stream talking head generation approaches, the 2D-based
+methods usually depend on generative adversarial networks
+(GANs) [6, 10, 16, 22, 28] for audio-to-lip mapping, and
+1
+arXiv:2301.03786v1  [cs.CV]  10 Jan 2023
+
+most of them perform competently on model generalization.
+However, since GANs need to simultaneously optimize a
+generator and a discriminator, the training process lacks sta-
+bility and is prone to mode collapse [11]. Due to this re-
+striction, the generated talking videos are of limited image
+quality, and difficult to scale to higher resolutions. By con-
+trast, 3D-based methods [2,17,42,46,53] perform better in
+synthesizing higher-quality talking videos. Whereas, they
+highly rely on identity-specific training, and thus cannot
+generalize across different persons. Such identity-specific
+training also brings heavy resource consumption and is not
+friendly to practical applications. Most recently, there are
+some 3D-based works [36] that take a step towards improv-
+ing the generalization of the model. However, further fine-
+tuning on specific identities is still inevitable.
+Generation quality and model generalization are two es-
+sential factors for better deployment of the talking head syn-
+thesis technique to real-world applications. However, few
+existing works are able to address both issues well. In this
+paper, we propose a crafted conditional Diffusion model for
+generalized Talking head synthesis (DiffTalk), that aims to
+tackle these two challenges simultaneously. Specifically, to
+avoid the unstable training of GANs, we turn attention to
+the recently developed generative technology Latent Dif-
+fusion Models [30], and model the talking head synthe-
+sis as an audio-driven temporally coherent denoising pro-
+cess. On this basis, instead of utilizing audio signals as
+the single driving factor to learn the audio-to-lip transla-
+tion, we further incorporate reference face images and land-
+marks as supplementary conditions to guide the face iden-
+tity and head pose for personality-aware video synthesis.
+Under these designs, the talking head generation process
+is more controllable, which enables the learned model to
+naturally generalize across different identities without fur-
+ther fine-tuning. As shown in Figure 1, with a sequence
+of driven audio, our DiffTalk is capable of producing natu-
+ral talking videos of different identities based on the corre-
+sponding reference videos. Moreover, benefiting from the
+latent space learning mode, our DiffTalk can be gracefully
+tailored for higher-resolution synthesis with negligible ex-
+tra computational cost, which is meaningful for improving
+the generation quality.
+Extensive experiments show that our DiffTalk can syn-
+thesize high-fidelity talking videos for novel identities with-
+out any further fine-tuning. Figure 1 shows the generated
+talking sequences with one driven audio across three differ-
+ent identities. Comprehensive method comparisons show
+the superiority of the proposed DiffTalk, which provides a
+strong baseline for the high-performance talking head syn-
+thesis. To summarize, we make the following contributions:
+• We propose a crafted conditional diffusion model for
+high-quality and generalized talking head synthesis. By
+introducing smooth audio signals as a condition, we
+model the generation as an audio-driven temporally co-
+herent denoising process.
+• For personality-aware generalized synthesis, we further
+incorporate dual reference images as conditions. In this
+way, the trained model can be generalized across different
+identities without further fine-tuning.
+• The proposed DiffTalk can generate high-fidelity and
+vivid talking videos for generalized identities. In exper-
+iment, our DiffTalk significantly outperforms 2D-based
+methods in the generated image quality, while surpassing
+3D-based works in the model generalization ability.
+2. Related Work
+Audio-driven Talking Head Synthesis.
+The talking
+head synthesis aims to generate talking videos with lip
+movements synchronized with the driving audio [14, 40].
+In terms of the modeling approach, we roughly divide the
+existing methods into 2D-based and 3D-based ones.
+In
+the 2D-based methods, GANs [6, 10, 16, 28] are usually
+employed as the core technologies for learning the audio-
+to-lip translation.
+Zhou et al. [52] introduce a speaker-
+aware audio encoder for personalized head motion model-
+ing. Prajwal et al. [28] boost the lip-visual synchroniza-
+tion with a well-trained Lip-Sync expert [8].
+However,
+since the training process of GANs lacks stability and is
+prone to mode collapse [11], the generated talking videos
+are always of limited image quality, and difficult to scale
+to higher resolutions. Recently a series of 3D-based meth-
+ods [4,20,39–41] have been developed. [39–41] utilize 3D
+Morphable Models [2] for parametric control of the talk-
+ing face.
+More recently, the emerging Neural radiance
+fields [26] provide a new solution for 3D-aware talking head
+synthesis [3, 17, 24, 36]. However, most of these 3D-based
+works highly rely on identity-specific training, and thus
+cannot generalize across different identities. Shen et al. [36]
+have tried to improve the generalization of the model, how-
+ever, further fine-tuning on specific identities is still in-
+evitable. In this work, we propose a brand-new diffusion
+model-based framework for high-fidelity and generalized
+talking head synthesis.
+Latent Diffusion Models. Diffusion Probabilistic Mod-
+els (DM) [37] have shown strong ability in various im-
+age generation tasks [11, 19, 29]. However, due to pixel
+space-based training [30,32], very high computational costs
+are inevitable.
+More recently, Rombach et al. [30] pro-
+pose the Latent Diffusion Models (LDMs), and transfer the
+training and inference processes of DM to a compressed
+lower-dimension latent space for more efficient comput-
+ing [13, 49]. With the democratizing of this technology, it
+has been successfully employed in a series of works, in-
+cluding text-to-image translation [21, 31, 33], super resolu-
+tion [7, 12, 27], image inpainting [23, 25], motion genera-
+tion [35,48], 3D-aware prediction [1,34,43]. In this work,
+2
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+������
+…
+������−1
+���1
+���
+������
+������
+������
+Reference
+Audio
+Landmark
+������
+concatenate
+concatenate
+���
+������
+������−1
+������
+…
+…
+Conditions
+0
+Diffusion Process
+Denoising Process
+������
+������
+���
+���
+Figure 2. Overview of the proposed DiffTalk for generalized talking head video synthesis. Apart from the audio signal condition to drive
+the lip motions, we further incorporate reference images and facial landmarks as extra driving factors for personalized facial modeling.
+In this way, the talking head generation process is more controllable, which enables the learned model to generalize across different
+identities without further fine-tuning. Furthermore, benefiting from the latent space learning mode, we can graceful improve our DiffTalk
+for higher-resolution synthesis with slight extra computational cost.
+drawing on these successful practices, we model the talk-
+ing head synthesis as an audio-driven temporally coherent
+denoising process and achieve superior generation results.
+3. Methodology
+3.1. Overview
+To tackle the challenges of generation quality and model
+generalization for better real-world deployment, we model
+the talking head synthesis as an audio-driven temporally co-
+herent denoising process, and term the proposed method as
+DiffTalk. An overview of the proposed DiffTalk is shown in
+Figure 2. By introducing smooth audio features as a condi-
+tion, we improve the diffusion model for temporally coher-
+ent facial motion modeling. For further personalized facial
+modeling, we incorporate reference face images and facial
+landmarks as extra driving factors. In this way, the talking
+head generation process is more controllable, which enables
+the learned model to generalize across different identities
+without any further fine-tuning. Moreover, benefiting from
+the latent space learning mode, we can graceful improve
+our DiffTalk for higher-resolution synthesis with negligible
+extra computational cost, which contributes to improving
+the generation quality. In the following, we will detail the
+proposed conditional Diffusion Models for high-fidelity and
+generalized talking head generation in Section 3.2. In Sec-
+tion 3.3, the progressive inference stage is introduced for
+better inter-frame consistency.
+3.2. Conditional Diffusion Model for Talking Head
+The emergence of Latent Diffusion Models (LDMs) [19,
+30] provides a straightforward and effective way for high-
+fidelity image synthesis. To inherit its excellent properties,
+we adopt this advanced technology as the foundation of our
+method and explore its potential in modeling the dynamic
+talking head. With a pair of well-trained image encoder EI
+and decoder DI which are frozen in training [13], the in-
+put face image x ∈ RH×W ×3 can be encoded into a latent
+space z0 = EI(x) ∈ Rh×w×3, where H/h = W/w = f,
+H, W are the height and width of the original image and
+f is the downsampling factor. In this way, the learning is
+transferred to a lower-dimensional latent space, which is
+more efficient with fewer train resources. On this basis, the
+standard LDMs are modeled as a time-conditional UNet-
+based [32] denoising network M, which learns the reverse
+process of a Markov Chain [15] of length T. The corre-
+sponding objective can be formulated as:
+LLDM := Ez,ϵ∼N (0,1),t
+�
+∥ϵ − M (zt, t)∥2
+2
+�
+,
+(1)
+where t ∈ [1, · · · , T] and zt is obtained through the forward
+diffusion process from z0. ˜zt−1 = zt − M(zt, t) is the
+denoising result of zt at time step t. The final denoised
+result ˜z0 is then upsampled to the pixel space with the pre-
+trained image decoder ˜x = DI(˜z0), where ˜x ∈ RH×W ×3
+is the reconstructed face image.
+Given a source identity and driven audio, our goal is to
+train a model for generating a natural target talking video in
+3
+
+Audio Stream
+16 time intervals
+DeepSpeech RNN
+Feature Map
+Feature 
+Extractor
+Temporal 
+Filtering
+16 windown size 
+Figure 3. Visualization of the smooth audio feature extractor. For
+better temporal coherence, two-stage smoothing operations are in-
+volved in this module.
+synchronization with the audio condition while maintaining
+the original identity information. Furthermore, the trained
+model also needs to work for novel identities during infer-
+ence. To this end, the audio signal is introduced as a basic
+condition to guide the direction of the denoising process for
+modeling the audio-to-lip translation.
+Smooth Audio Feature Extraction. To better incorpo-
+rate temporal information, we involve two-stage smoothing
+operations in the audio encoder EA, as shown in Figure 3.
+Firstly, following the practice in VOCA [9], we reorganize
+the raw audio signal into overlapped windows of size 16
+time intervals (corresponding to audio clips of 20ms), where
+each window is centered on the corresponding video frame.
+A pre-trained RNN-based DeepSpeech [18] module is then
+leveraged to extract the per-frame audio feature map F. For
+better inter-frame consistency, we further introduce a learn-
+able temporal filtering [41]. It receives a sequence of adja-
+cent audio features [Fi−w, . . . , Fi, . . . , Fi+w] with w = 8
+as input, and computes the final smoothed audio feature for
+the i-th frame as a ∈ RDA in a self-attention-based learn-
+ing manner, where DA denotes the audio feature dimension.
+By encoding the audio information, we bridge the modality
+gap between the audio signals and the visual information.
+Introducing such smooth audio features as a condition, we
+extend the diffusion model for temporal coherence-aware
+modeling of face dynamics when talking. The objective is
+then formulated as:
+LA := Ez,ϵ∼N (0,1),a,t
+�
+∥ϵ − M (zt, t, a)∥2
+2
+�
+.
+(2)
+Identity-Preserving Model Generalization. In addi-
+tion to learning the audio-to-lip translation, another essen-
+tial task is to realize the model generalization while pre-
+serving complete identity information in the source image.
+Generalized identity information includes face appearance,
+head pose, and image background. To this end, a reference
+mechanism is designed to empower our model to general-
+ize to new individuals unseen in training, as shown in Fig-
+ure 2. Specifically, a random face image xr of the source
+identity is chosen as a reference condition, which contains
+appearance and background information. To prevent train-
+ing shortcuts, we limit the selection of xr to 60 frames be-
+yond the target image.
+However, since the ground-truth
+face image has a completely different pose from xr, the
+model is expected to transfer the pose of xr to the target
+face without any prior information. This is somehow an
+ill-posed problem with no unique solution. For this rea-
+son, we further incorporate the masked ground-truth im-
+age xm as another reference condition to provide the target
+head pose guidance. The mouth region of xm is completely
+masked to ensure that the ground truth lip movements are
+not visible to the network. In this way, the reference xr fo-
+cuses on affording mouth appearance information, which
+additionally reduces the training difficulty.
+Before serv-
+ing as conditions, xr and xm are also encoded into the la-
+tent space through the trained image encoder, and we have
+zr = DI(xr) ∈ Rh×w×3, zm = DI(xm) ∈ Rh×w×3. On
+this basis, an auxiliary facial landmarks condition is also in-
+cluded for better control of the face outline. Similarly, land-
+marks in the mouth area are masked to avoid shortcuts. The
+landmark feature l ∈ RDL is obtained with an MLP-based
+encoder EL, where DL is the landmark feature dimension.
+In this way, combining these conditions with audio feature
+a, we realize the precise control over all key elements of
+a dynamic talking face. With C = {a, zr, zm, l} denoting
+the condition set, the talking head synthesis is finally mod-
+eled as a conditional denoising process optimized with the
+following objective:
+L := Ez,ϵ∼N (0,1),C,t
+�
+∥ϵ − M (zt, t, C)∥2
+2
+�
+,
+(3)
+where the network parameters of M, EA and EL are jointly
+optimized via this equation.
+Conditioning Mechanisms. Based on the modeling of
+the conditional denoising process in Eq. 3, we pass these
+conditions C to the network in the manner shown in Fig-
+ure 2. Specifically, following [30], we implement the UNet-
+based backbone M with the cross-attention mechanism for
+better multimodality learning. The spatially aligned refer-
+ences zr and zm are concatenated channel-wise with the
+noisy map zT to produce a joint visual condition Cv =
+[zT ; zm; zr] ∈ Rh×w×9. Cv is fed to the first layer of the
+network to directly guide the output face in an image-to-
+image translation fashion. Additionally, the driven-audio
+feature a and the landmark representation l are concatenated
+into a latent condition Cl = [a; l] ∈ RDA+DL, which serves
+as the key and value for the intermediate cross-attention
+layers of M.
+To this extent, all condition information
+C = {Cv, Cl} are properly integrated into the denoising
+network M to guide the talking head generation process.
+4
+
+DDIM-based Denoising
+������,1
+Random ������,1
+���1
+������,2
+������,2
+���2
+������
+DDIM-based Denoising
+DDIM-based Denoising
+������,���
+…
+������,���
+Figure 4. Illustration of the designed progressive inference strat-
+egy. For the first frame, the setting of the visual condition Cv
+remains the same as for training, where xr,1 is a random face im-
+age from the target identity. Subsequently, the synthetic image ˜xi
+is employed as the reference condition xr,i+1 for the next frame
+to enhance the temporal coherence of the generated video.
+Higher-Resolution Talking Head Synthesis Our pro-
+posed DiffTalk can also be gracefully extended for higher-
+resolution talking head synthesis with negligible extra com-
+putational cost and faithful reconstruction effects. Specif-
+ically, considering the trade-off between the perceptual
+loss and the compression rate, for training images of size
+256 × 256 × 3, we set the downsampling factor as f = 4
+and obtain the latent space of 64 × 64 × 3. Furthermore,
+for higher-resolution generation of 512 × 512 × 3, we just
+need to adjust the paired image encoder EI and decoder DI
+with a bigger downsampling factor f = 8. Then the trained
+encoder is frozen and employed to transfer the training pro-
+cess to a 64 × 64 × 3 latent space as well. This helps to
+relieve the pressure on insufficient resources, and therefore
+our model can be gracefully improved for higher-resolution
+talking head video synthesis.
+3.3. Progressive Inference
+We perform inference with Denoising Diffusion Implicit
+Model-based (DDIM) [38] iterative denoising steps. DDIM
+is a variant of the standard DM to accelerate sampling for
+more efficient synthesis. To further boost the coherence of
+the generated talking videos, we develop a progressive ref-
+erence strategy in the reference process as shown in Fig-
+ure 4. Specifically, when rendering a talking video sequence
+with the trained model, for the first frame, the setting of the
+visual condition Cv remains the same as for training, where
+xr,1 is a random face image from the target identity. Sub-
+sequently, this synthetic face image is exploited as the xr
+for the next frame. In this way, image details between adja-
+cent frames remain consistent, resulting in a smoother tran-
+sition between frames. It is worth noting that this strategy
+is not used for training. Since the difference between adja-
+cent frames is small, we need to eliminate such references
+to avoid learning shortcuts.
+0
+100
+0
+100
+GT
+w.o. Smooth
+w. Smooth
+Figure 5. Ablation study on the audio smoothing operation. We
+show the differences between adjacent frames as heatmaps for bet-
+ter visualization. The results without audio filtering present obvi-
+ous high heat values in the mouth region, which indicates the jitters
+in this area. By contrast, with smooth audio as the condition, the
+generated video frames show smoother transitions.
+4. Experiments
+4.1. Experimental Settings
+Dataset. To train the audio-driven diffusion model, an
+audio-visual dataset HDTF [51] is used.
+It contains 16
+hours of talking videos in 720P or 1080P from more than
+300 identities.
+We randomly select 100 videos with the
+length of about 5 hours for training, while the remaining
+data serve as the test set. Apart from this public dataset, we
+also use some other videos for cross-dataset evaluation.
+Metric. We evaluate our proposed method through vi-
+sual results coupled with quantitative indicators.
+PSNR
+(↑), SSIM (↑) [45] and LPIPS (↓) [49] are three metrics
+for assessing image quality. The LPIPS is a learning-based
+perceptual similarity measure that is more in line with hu-
+man perception, we therefore recommend this metric as a
+more objective indicator.
+The SyncNet score (Offset↓ /
+Confidence↑) [8] checks the audio-visual synchronization
+quality, which is important for the audio-driven talking head
+generation task.
+(‘↓’ indicates that the lower the better,
+while ‘↑’ means that the higher the better.)
+Implementation Details. We resize the input image to
+256 × 256 for experiments. The downsampling factor f is
+set as 4, so the latent space is 64 × 64 × 3. For training the
+model for higher resolution synthesis, the input is resized to
+512 × 512 with f = 8 to keep the same size of latent space.
+The length of the denoising step T is set as 200 for both the
+5
+
+Ground Truth
+A
+A + L
+A + L + R
+A + M
+A + L + M + R
+ID 1
+ID 2
+Figure 6. Ablation study on the design of the conditions. The marks above these images refer to the following meanings, ‘A’: Audio;
+‘L’: Landmark; ‘R’: Random reference image; ‘M’: Masked ground-truth image. We show the generated results under different condition
+settings on two test sets, and demonstrate the effectiveness of our final design, i.e. A+L+M+R.
+Method
+PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑
+Test Set A
+GT
+-
+-
+-
+0/9.610
+w/o
+33.67
+0.944
+0.024
+1/5.484
+w
+34.17
+0.946
+0.024
+1/6.287
+Test Set B
+GT
+-
+-
+-
+0/9.553
+w/o
+32.70
+0.924
+0.031
+1/5.197
+w
+32.73
+0.925
+0.031
+1/5.387
+Table 1. Ablation study to investigate the contribution of the audio
+smoothing operation. ‘w’ indicates the model is trained with the
+audio features after temporal filtering and vice versa.
+training and inference process. The feature dimensions are
+DA = DL = 64. Our model takes about 15 hours to train
+on 8 NVIDIA 3090Ti GPUs.
+4.2. Ablation Study
+Effect of the Smooth Audio. In this subsection, we in-
+vestigate the effect of the audio smooth operations. Quanti-
+tative results in Table 1 show that the model equipped with
+the audio temporal filtering module outperforms the one
+without smooth audio, especially in the SyncNet score. We
+further visualize the differences between adjacent frames as
+the heatmaps shown in Figure 5. The results without audio
+filtering present obvious high heat values in the mouth re-
+gion, which indicates the jitters in this area. By contrast,
+with smooth audio as the condition, the generated video
+frames show smoother transitions, which are reflected in the
+soft differences of adjacent frames.
+Design of the Conditions. A major contribution of this
+work is the ingenious design of the conditions for general
+and high-fidelity talking head synthesis. In Figure 6, we
+show the generated results under different condition settings
+step by step, to demonstrate the superiority of our design.
+Method
+PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑
+Test Set A
+GT
+-
+-
+-
+4/7.762
+w/o
+34.17
+0.946
+0.024
+1/6.287
+w
+33.95
+0.946
+0.023
+-1/6.662
+Test Set B
+GT
+-
+-
+-
+3/8.947
+w/o
+32.73
+0.925
+0.031
+1/5.387
+w
+33.02
+0.925
+0.030
+1/5.999
+Table 2. Ablation study on the effect of the progressive inference
+strategy. ‘w/o’ indicates that a random reference image is em-
+ployed as the condition, and ‘w’ means that the reference is the
+generated result of the previous frame.
+With pure audio as the condition, the model fails to gener-
+alize to new identities, and the faces are not aligned with the
+background in the inpainting-based inference. Adding land-
+marks as another condition tackles the misalignment prob-
+lem. A random reference image is further introduced try-
+ing to provide the identity information. Whereas, since the
+ground-truth face image has a different pose from this ran-
+dom reference, the model is expected to transfer the pose of
+reference to the target face. This greatly increases the diffi-
+culty of training, leading to hard network convergence, and
+the identity information is not well learned. Using the au-
+dio and masked ground-truth images as driving factors mit-
+igates the identity inconsistency and misalignment issues,
+however the appearance of the mouth can not be learned
+since this information is not visible to the network. For
+this reason, we employ the random reference face and the
+masked ground-truth image together for dual driving, where
+the random reference provides the lip appearance message
+and the masked ground-truth controls the head pose and
+identity. Facial landmarks are also incorporated as a con-
+dition that helps to model the facial contour better. Results
+6
+
+GT
+ATVG
+MakeItTalk
+Wav2Lip
+Ours
+DFRF
+AD-NeRF
+3D-based Methods
+2D-based Methods
+Figure 7. Visual comparison with some representative 2D-based talking head generation methods ATVGnet [5], MakeitTalk [52] and
+Wav2Lip [28], and with some recent 3D-based ones AD-NeRF [17] and DFRF [36]. The results of DFRF are synthesized with the base
+model without fine-tuning for fair comparisons. AD-NeRF is trained on these two identities respectively to produce the results.
+in Figure 6 show the effectiveness of such design in synthe-
+sizing realism and controllable face images.
+Impact of the Progressive Inference. Temporal corre-
+lation inference is developed in this work through the pro-
+gressive reference strategy. We conduct an ablation study
+in Table 2 to investigate the impact of this design. ‘w/o’ in-
+dicates that a random reference image xr is employed, and
+‘w’ means that the generated result of the previous frame
+is chosen as the reference condition. With such progressive
+inference, the SyncNet scores are further boosted, since the
+temporal correlation is better modeled and the talking style
+becomes more coherent. The LPIPS indicator is also en-
+hanced with this improvement. PSNR tends to give higher
+scores to blurry images [49], so we recommend LPIPS as a
+more representative metric for visual quality.
+4.3. Method Comparison
+Comparison with 2D-based Methods. In this section,
+we perform method comparisons with some representative
+2D-based talking head generation approaches including the
+ATVGnet [5], MakeitTalk [52] and Wav2Lip [28]. Figure 7
+visualizes the generated frames of these methods. It can
+be seen that the ATVGnet performs generation based on
+cropped faces with limited image quality. The MakeItTalk
+synthesizes plausible talking frames, however the back-
+ground is wrongly wrapped with the mouth movements.
+This phenomenon is more observable in the video form
+result, and greatly affects the visual experience.
+Gener-
+ated talking faces of Wav2Lip appear artifacts in the square
+boundary centered on the mouth, since the synthesized area
+7
+
+Method
+Test Set A
+Test Set B
+General
+PSNR↑
+SSIM↑
+LPIPS↓
+SyncNet↓↑
+PSNR↑
+SSIM↑
+LPIPS↓
+SyncNet↓↑
+Method
+GT
+-
+-
+-
+-1/8.979
+-
+-
+-
+-2/7.924
+-
+MakeItTalk [52]
+18.77
+0.544
+0.19
+-4/3.936
+17.70
+0.648
+0.129
+-3/3.416
+✓
+Wav2Lip [28]
+25.50
+0.761
+0.140
+-2/8.936
+33.38
+0.942
+0.027
+-3/9.385
+✓
+AD-NeRF [17]
+27.89
+0.885
+0.072
+-2/5.639
+30.14
+0.947
+0.023
+-3/4.246
+
+DFRF [36]
+28.60
+0.892
+0.068
+-1/5.999
+33.57
+0.949
+0.025
+-2/4.432
+FT Req.
+Ours
+34.54
+0.950
+0.024
+-1/6.381
+34.01
+0.950
+0.020
+-1/5.639
+✓
+Table 3. Comparison with some representative talking head synthesis methods on two test sets as in Figure 7. The best performance is
+highlighted in red (1st best) and blue (2nd best). Our DiffTalk obtains the best PSNR, SSIM, and LPIPS values, and comparable SyncNet
+scores simultaneously. It is worth noting that the DFRF is fine-tuned on the specific identity to obtain these results, while our method can
+directly be utilized for generation without further fine-tuning. (‘FT Req.’ means that fine-tuning operation is required for the DFRF.)
+and the original image are not well blended. By contrast,
+the proposed DiffTalk generates natural and realistic talk-
+ing videos with accurate audio-lip synchronization, owing
+to the crafted conditioning mechanism and stable training
+process. For more objective comparisons, we further eval-
+uate the quantitative results in Table 3. Our DiffTalk far
+surpasses [28] and [52] in all image quality metrics. For
+the audio-visual synchronization metric SyncNet, the pro-
+posed method reaches a high level and is superior than
+MakeItTalk. Although the DiffTalk is slightly inferior to
+Wav2Lip on the SyncNet score, it is far better than Wav2Lip
+in terms of image quality. In conclusion, our method outper-
+forms these 2D-based methods under comprehensive con-
+sideration of the qualitative and quantitative results.
+Comparison with 3D-based Methods. For more com-
+prehensive evaluations, we further compare with some
+recent high-performance 3D-based works including AD-
+NeRF [17] and DFRF [36]. They realize implicitly 3D head
+modeling with the NeRF technology, so we treat them as
+generalized 3D-based methods. The visualization results
+are shown in Figure 7. AD-NeRF models the head and torso
+parts separately, resulting in misalignment in the neck re-
+gion. More importantly, it is worth noting that AD-NeRF
+is a non-general method. In contrast, our method is able to
+handle unseen identities without further fine-tuning, which
+is more in line with the practical application scenarios. The
+DFRF relies heavily on the fine-tuning operation for model
+generalization, and the generated talking faces with only
+the base model are far from satisfactory as shown in Fig-
+ure 7. More quantitative results in Table 3 also show that our
+method surpasses [17, 36] on the image quality and audio-
+visual synchronization indicators.
+4.4. Expand to Higher Resolution
+In this section, we perform experiments to demonstrate
+the capacity of our method on generating higher-resolution
+images. In Figure 8, we show the synthesis frames of two
+models (a) and (b). (a) is trained on 256 × 256 images with
+the downsampling factor f = 4, so the latent space is of
+(a) Resolution: 256 × 256, ���=4
+(b) Resolution: 512 × 512,  ���=8
+Figure 8. Generated results with higher resolution.
+size 64 × 64 × 3. For (b), 512 × 512 images with f =
+8 are used for training the model. Since both models are
+trained based on a compressed 64 × 63 × 3 latent space,
+the pressure of insufficient computing resources is relieved.
+We can therefore comfortably expand our model for higher-
+resolution generation just as shown in Figure 8, where the
+synthesis quality in (b) significantly outperforms that in (a).
+5. Conclusion and Discussion
+In this paper, we have proposed a generalized and high-
+fidelity talking head synthesis method based on a crafted
+conditional diffusion model. Apart from the audio signal
+condition to drive the lip motions, we further incorporate
+reference images as driving factors to model the personal-
+ized appearance, which enables the learned model to com-
+fortably generalize across different identities without any
+further fine-tuning. Furthermore, our proposed DiffTalk can
+be gracefully tailored for higher-resolution synthesis with
+negligible extra computational cost.
+Limitations. The proposed method models talking head
+generation as an iterative denoising process, which needs
+more time to synthesize a frame compared with most GAN-
+based approaches. This is also a common problem of LDM-
+based works which warrants further research. Nonetheless,
+we have a large speed advantage over most 3D-based meth-
+ods. Since talking head technology may raise potential mis-
+use issues, we are committed to combating these malicious
+behaviors and advocate positive applications. Additionally,
+researchers who want to use our code will be required to get
+authorization and add watermarks to the generated videos.
+8
+
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