Interventionally Consistent Surrogates for Complex Simulation Models

Authors: joel  dyer
nicholas  bishop
yorgos  felekis
fabio massimo zennaro
anisoara  calinescu
theodoros  damoulas
michael  wooldridge
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 19:50:47

Abstract: Large-scale simulation models of complex socio-technical systems provide decision-makers with high-fidelity testbeds in which policy interventions can be evaluated and _what-if_ scenarios explored. Unfortunately, the high computational cost of such models inhibits their widespread use in policy-making settings. Surrogate models can address these computational limitations, but to do so they must behave consistently with the simulator under interventions of interest. In this paper, we build upon recent developments in causal abstractions to develop a framework for learning interventionally consistent surrogate models for large-scale, complex simulation models. We provide theoretical results showing that our proposed approach induces surrogates to behave consistently with high probability with respect to the simulator across interventions of interest, facilitating rapid experimentation with policy interventions in complex systems. We further demonstrate with empirical studies that conventionally trained surrogates can misjudge the effect of interventions and misguide decision-makers towards suboptimal interventions, while surrogates trained for _interventional_ consistency with our method closely mimic the behaviour of the original simulator under interventions of interest.

Main paper: 
 introduction: 
 Large-scale, complex simulators are powerful tools for modelling distributed socio-technical systems and emergent phenomena across application domains, including the social sciences [Wiese et al., 2024] , epidemiology [Kerr et al., 2021] , and finance [Cont, 2007] . Many such systems consist of a multitude of autonomous, interacting, and decision-making agents, whose individual behaviours and interactions can be captured more readily and at a higher degree of fidelity in a computer program than through conventional modelling paradigms. This level of granularity can, in turn, allow for more effective control of the potentially deleterious effects that can arise from the endogenous dynamics of real-world systems by providing a testbed for experimentation with policy interventions. In economics, for example, such interventions may take the form of limits on loan-to-value ratios in housing markets to attenuate housing price cycles [Baptista et al., 2016] , while in epidemiology they may be (non-)pharmaceutical interventions that aim to inhibit disease transmission [Kerr et al., 2021] .Whilst simulation modelling of this kind promises many benefits, the intricacy and multi-scale nature of the simulators that result from these modelling efforts can result in large computational costs even for single forward simulations [Jagiella et al., 2017 , Fadai et al., 2019 , Wright and Davidson, 2020 , Heppenstall et al., 2021] . Since extensive simulation studies are often required to aid decision-making with these models, such costs present a barrier to their use as synthetic test environments for potential policy interventions in practice. Moreover, the high-fidelity data generated by detailed simulation models can be difficult for decision-makers to interpret and relate to policy interventions that act system-wide [Haldane and Turrell, 2018] . This motivates the development of simpler surrogate models that model the underlying system at a higher level of abstraction. Such surrogates can also be used in place of the complex model for downstream tasks where computational resources are limited. In addition, surrogates may be viewed as interpretable explanations for the complex simulator, and they allow for rapid testing of population-wide interventions which may be difficult to implement or test within the orginal model.However, for surrogates to be useful in downstream tasks involving experimentation with possible policy interventions, they must preserve the complex simulator's dynamics under the external interventions of interest. Without imposing this condition on the constructed surrogate, there is no guarantee that the surrogate will behave similarly under external policy interventions, which in turn may lead policy-makers away from effective policies and towards suboptimal interventions. Existing methods typically apply off-the-shelf machine learning methods to learn surrogates through observation [Lamperti et al., 2018 , Platt, 2022] , which fails to account for interventional consistency.Our contribution. To address this, we build on recent developments in causal abstraction [Beckers and Halpern, 2019, Zennaro et al., 2023a] . We view the complex simulator and its surrogate as structural causal models [Pearl, 2009] , and propose a framework for constructing and learning surrogate models for expensive simulators of complex socio-technical systems that are interventionally consistent, in the sense that they (approximately) preserve the behaviour of the simulator under equivalent policy interventions. This perspective enables treating the surrogate model as a causal abstraction of the simulator. We motivate our proposed methodology theoretically, and demonstrate with simulation studies that our method permits us to learn an abstracted surrogate model for an epidemiological agent-based model that behaves consistently in multiple interventional regimes.Our approach establishes, for the first time, a connection between complex simulation models and causal abstraction, and a practical approach to learning interventionally consistent surrogates for complex simulators. Our work provides an avenue for researchers modelling complex socio-technical systems to draw on the rich literature in causality for integrating causal knowledge, evaluating what-if scenarios, and learning new abstracted models with guarantees about interventional consistency. Our contribution lays the groundwork for surrogate modelling methods that facilitate rapid experimentation with different scenarios and interventions, with assurances that the error introduced by experimenting at a higher level of abstraction is low. This line of work has the potential to enable decision-and policy-makers to use simulation models to quickly identify life-saving policy strategies during novel and rapidly unfolding emergencies, such as pandemics and economic crises. Indeed, a recent World Health Organisation report [Health Organization et al., 2024] emphasises the importance of integrated modelling to concurrently address interdependent policy objectives, such as reducing disease transmission, mitigating hospital admissions overload, and minimising the economic costs of service closures on society during pandemics. It further discusses the intense time pressures involved in these efforts. Our work addresses these points by taking steps towards facilitating rapid experimentation with large and computationally expensive integrated simulation models. 
background: 
 We first recall the key elements of causal inference, following Pearl [2009] , and elucidate the connection between structural causal models (SCMs) and complex simulators. We also review the notion of exact transformations between SCMs, which theoretically motivates our framework.A SCM is a rigorous model describing a causal system: Definition 1 (SCMs [Pearl, 2009] ). A structural causal model M is a tuple ⟨X, U, F, P(U)⟩ where:• X = {X i } ni=1 , is a finite set of endogenous random variables X i each with domain dom[X i ];• U = {U i } n i=1 , is a finite set of exogenous random variables, each with domain dom[U i ] and each associated with an endogenous variable;• F = {f i } ni=1 , is a finite set of measurable structural functions, one for each endogenous variable defined as f i : dom[P A(X i )] × dom[U i ] → dom[X i ], where P A(X i ) ⊆ X \ X i .• P M (U) is a joint probability distribution over the exogenous variables factorizing asn i=1 P M (U i ).The model M is associated with a Directed Acyclic Graph (DAG) G M = ⟨V, E⟩ where the set V of vertices is given by X ∪ U and the set E of edges is given by {(S j , X i ) | S j ∈ P A(X i ) ∪ {U i }} n i=1 . Definition 1 conforms to the standard definition of a Markovian SCM (see Appendix A for an explanation of the underlying assumptions). Thanks to the measurability of the structural functions in F, the probability distribution P M (U) over the exogenous variables can be pushed forward over the endogenous variables, defining the probability distribution P M (X) = F # (P M (U)). Joint distributions P M (S) can then be defined for any subset S ⊆ X.External interventions on the system by an experimenter can be represented in an SCM through changes in the structural functions. Here, we restrict our attention to hard interventions, in which fixed values are assigned to subsets of endogenous variables: Definition 2 (Interventions [Pearl, 2009] ). Given an SCM M, S ⊆ X, and a set of values s realizing S, an intervention ι = do(S = s), is an operator that replaces each function f i associated with S i with constant s i .The intervention ι = do(S = s) induces a new post-intervention SCM, M ι = ⟨X, U, F ι , P(U)⟩, identical to the original one, except that in F ι the functions f i are replaced with the constants s i . The probability distribution of M ι is computed as P Mι (X \ S). Graphically, the intervention ι transforms the DAG of M by removing incoming edges in each variable S i .We use I to denote a set of feasible interventions on the SCM M that are relevant to a policymaker. Intervention sets are equipped with a natural partial ordering: let ι 1 = (S = s) and ι 2 = (T = t); then ι 1 ⪯ ι 2 iff (i) S ⊆ T, and (ii) for each S i = T i it holds s i = t i ; informally, ι 1 intervenes on a subset of the variables that ι 2 intervenes on, and it sets the same values as ι 2 .Many simulation models of complex systems -such as, for example, agent-based models (ABMs) -can be modelled as a SCM by expressing its implicit underlying causal structure. Practically, this entails encoding quantities of interest as endogenous variables, deterministic dynamics into structural equations, and factoring sources of randomness into exogenous variables. The following example illustrates how a common ABM from epidemiology can be cast as a SCM. Example 1 (Spatial SIRS ABM). We consider a susceptible-infected-recovered-susceptible (SIRS) epidemic model on an L × L lattice of cells, each of which represents one of N = L 2 agents. The state of each agent can be 0, 1, or 2, respectively, indicating that the agent is disease-free and susceptible to infection, infected, or is recovered from a recent infection. The infection status of all agents at discrete time step t ∈ 0, T is written as x t ∈ {0, 1, 2} N , where T is the total number of simulated time steps, and l, m = {l, l + 1, . . . , m -1, m} for integers l ≤ m. The states x t,n of each of the agents n ∈ 1, N are updated synchronously as follows for t ∈ 0, T -1 :(U1) If x t,n = 0, then x t+1,n = 1 with probability p t,n (α t+1 ) = 1 -(1 -α t+1 ) n ′ ∈Nn I[x t,n ′ =1](1) where N n is the von Neumann neighbourhood for cell n; else remain susceptible.(U2) If x t,n = 1, then x t+1,n = 2 with probability β t+1 ; else remain infected.(U3) If x t,n = 2, then x t+1,n = 0 with probability γ t+1 ; else remain recovered.In the above, θ t = (α t , β t , γ t ) ∈ [0, 1] 3 are the model parameters determining the transition probabilities between states. While these may vary over time, the simplest case consists of assigning all θ t the same vector,EQUATION)The model is initialised by infecting each agent in the model at initial time t = 0 with probability I 0 ∈ [0, 1]. The value of I 0 for any forward simulation of the model can be chosen by drawing a random variable a from some distribution on [0, 1] and settingEQUATIONWith this model in place, lockdowns over some time period t l : t l + ∆ of length ∆ ≥ 0 can be modelled (crudely) by setting θ t l :t l +∆ = (0, β, γ) for β, γ ∈ [0, 1]. To express this ABM as an SCM, we define the following:Endogenous variables These consist of the variables of interest that may be set by the policymaker: I 0 , {x t } 0≤t≤T , and {θ t } 1≤t≤T .The model as described above is initialised randomly according to a, v, and a collection u 0 = (u 0,n ) 1≤n≤N of N random variables distributed as U(0, 1), the nth of which helps determine whether agent n is infected at time t = 0. Similarly, further collections u t , t ∈ 1, T of U(0, 1) random variables decide how each agent updates their state at each time step. Thus the exogenous variables for the model are a, v, and the u t for t ∈ 0, T .Structural equations Equations 2 and 3, respectively, define the structural equations f θt and f I0 for the endogenous variables θ t and I 0 . The structural equation f x0,n for each x 0,n , n ∈ 1, N can furthermore be written asEQUATIONFinally, update rules (U1)-(U3) can be written in the following way for t ∈ 0, T -1 :EQUATIONDistribution over exogenous variables The (random) behaviour of the exogenous variables is fully specified by the distribution over a and v, along with U(0, 1) distributions over the u t,n .The DAG corresponding to this SCM is shown in Figure 1 for T = 3.In this model, interventions in the form of, for example, lockdowns can be (crudely) modelled by intervening on one or more of the θ t as do(θ t = (0, β, γ)) for some β, γ ∈ [0, 1], while in the observational regime the θ t will all be assigned the same value. We emphasise that the above example is intended only to illustrate that complex simulators, such as ABMs, can be seen as SCMs; explicitly representing a given simulator as an SCM as in the example above is not required in the sequel.a I 0 x 0 x 1 x 2 x 3 u 0 u 1 u 2 u 3 θ 1 v θ 2 θ 3Beside expressing interventions more rigorously, viewing complex simulation models as SCMs allows one to take advantage of the theory of causal abstraction to formalise the relationship between the simulator and its surrogate model. Indeed, causal abstraction provides a framework for relating SCMs representing an identical system at different levels of granularity. The notion of exact transformation formalizes this relation, providing a framework to relate complex models, such as ABMs, to simpler top-down models, while preserving causal structure. Definition 3 (τ -ω Exact Transformation [Rubenstein et al., 2017] ). Given two SCMs, M and M ′ , with respective intervention sets I and I ′ , a τ -ω transformation is a pair (τ, ω) consisting of a map τ : dom[X] → dom[X ′ ] and a surjective, order-preserving map ω :EQUATIONAn exact τ -ω transformation constitutes a form of abstraction between probabilistic causal models [Beckers et al., 2020] with the guarantee of commutativity between intervention and transformation as detailed in Figure 2 : intervening via ι and then abstracting produces the same result as abstracting first and then intervening via ω(ι). The map τ describes corresponding states in each of the models, while the map ω describes corresponding interventions in each model. Whenever the map τ is clear from context, we herein shorthand the pushforward measure τ # (P M ι ) as P # Mι . ι P M ι ω(ι) P M ′ ω(ι) M ω τ M ′ Figure 2: Computing τ # (P M ι )which models how households decide to take precautions to protect themselves from floods in high flood risk areas. The modellers are interested in the different precautions households take under different policy interventions, namely whether they: do nothing; purchase insurance; purchase property-level protection; and purchase property-level protection and insurance (see Fig. 3 of Geaves et al. [2024] ). τ would count the number of agents taking such actions in this case (as in the example above).3. Consider the UK housing market model proposed in Bardoscia et al. [2024] , in which households consume goods, provide labour and invest in housing, whilst banks assess the credit worthiness of borrowers and set commercial interest rates. Tables 2-6 of Bardoscia et al. [2024] define macroeconomic market statistics such as inflation rate, unemployment rate and real interest rate that are of interest to the modellers. τ would therefore be defined by standard macroeconomic formulas for these quantities. : 
abstraction error: 
 It is often unrealistic to assume that an exact τ -ω transformation exists between a complex simulator and its surrogate A more pragmatic goal is to find an approximate abstraction [Beckers et al., 2020] from the simulator to the surrogate. We therefore define the abstraction error: Definition 4 (Abstraction error). Let (τ, ω) be a τ -ω transformation between two SCMs M and M ′ with respective intervention sets I and I ′ . Given a statistical divergence d between distributions, and a distribution η over the intervention set I, we define the abstraction error as follows:EQUATION) A τ -ω transformation is α-approximate for some α ∈ R ≥0 if d τ,ω (M, M ′ ) ≤ α. ι P M ι τ # (P M ι ) ω(ι) P M ′ ω(ι) M ω τ M ′ dτ,ω≤αFigure 3 : The abstraction error compares the distributions τ # (P M ι ) and P M ′ ω(ι) for each intervention ι using the divergence d τ,ω , as indicated by the red dotted arrow. If the divergence is zero then we recover the commutative diagram in Figure 2 .A τ -ω abstraction with low abstraction error implies that τ # (P M ι ) is close to P M ′ ω(ι) in expectation with respect to the interventional distribution η.If the d τ # (P M ι ), P M ′ ω(ι)is zero for all interventions ι ∈ I, then (τ, ω) is an exact transformation (see Figure 3 ). Definition 4 differs from previously defined notions of abstraction error in the causal abstraction literature. Whilst Beckers et al. [2020] employ a maximum over interventions, we instead take an expectation over a fixed interventional distribution η. This is motivated by the fact that policymakers will often hold preferences over possible interventions, which may, for example, reflect the cost or feasibility of implementing each intervention in the real world. Through the specification of η, one may implicitly favour surrogates which perform well with respect to interventions of high interest. Further discussion is provided in Appendix B.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] Justification: The assumptions are provided in the Propositions in Section 4.1 (and more generally in Section 2 and Appendix A), and the proofs are located in Appendices D.1 and D.2.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
method: 
 The definitions of abstraction and abstraction error provide us with a framework for learning surrogates, and, in the remainder, we assume that the base model M is implicitly represented by a simulation model of a complex socio-technical system. Our goal is then to identify a surrogate model which is interventionally consistent with this simulator. Specifically, from a set of candidate surrogate models M, we seek a surrogate and a τ -ω transformation that minimises the abstraction error.To proceed, we assume that M is a parameterised family M Ψ := {M ψ : ψ ∈ Ψ} of differentiable surrogate simulators with tractable probability mass or density function q ψ . Here, M ψ denotes the causal model induced by a surrogate whose structural equations are parameterised by ψ, and Ψ denotes the set of feasible parameter values. Such a family of surrogate models can be constructed through a composition of differential equation-or deep learning-based modelling, in combination with probability distributions with reparameterisable sampling procedures; an example is a latent neural ordinary differential equation model [Rubanova et al., 2019] , which we use in the experiments in Section 5. We further assume only the ability to sample from τ # (P M ), amounting to running the simulator and applying τ to the output.Generally speaking, policymakers know what macroscopic quantities are of interest when modelling a complex system, and how to aggregate the microscopic variables into global statistics. For example, in macroeconomic settings, policymakers will often be concerned with aggregate quantities such as unemployment rates or aggregate demand, which can be derived from the state of the agents. Further specific examples are discussed in Appendix C.1. We thus assume that the map τ , which defines the aggregate, emergent quantities of interest to the policymaker, is pre-specified.Hence, to find an appropriate τ -ω transformation, we need only to identify an intervention map ω ⋆ between I and I ′ . For computational tractability, we select ω ⋆ from a parameterised family Ω Φ := {ω ϕ : ϕ ∈ Φ} with parameters ϕ ranging over the set Φ. For example, ϕ may be the weights of a neural network. We then select ϕ ⋆ and ψ ⋆ jointly by minimising d τ,ω (M, M ′ ) over Ω Φ × M Ψ . Since each element of M has a differentiable and tractable distribution, a convenient choice of discrepancy d is the Kullback-Leibler (KL) divergence, such that out problem becomes:ϕ ⋆ , ψ ⋆ = arg min ϕ∈Φ,ψ∈Ψ d τ,ω ϕ (M, M ψ ) with d τ,ω ϕ (M, M ψ ) = E η E P # Mι   log dP # Mι dP M ψ ω ϕ (ι)   . (8)The KL divergence can be minimised using Monte Carlo estimates of the gradientG(ϕ, ψ) = ∇ ϕ,ψ d τ,ω ϕ (M, M ψ ) ≈ 1 B B b=1 -∇ ϕ,ψ log q ψ ω ϕ (ι (b) ) (y (b) ) (9)whereι (b) ∼ η, y (b) ∼ τ # (P M ι (b) ), q ψ ω ϕ (ι)is the probability mass/density function for M ψ ω ϕ (ι) , and B ≥ 1 is the size of a batch drawn from R ≥ B training examples from the joint distribution over the ι (b) and y (b) . Once (ϕ ⋆ , ψ ⋆ ) has been selected, we may generate data from the macromodel for ABM intervention ι by sampling fromP M ψ ⋆ ω ϕ ⋆ (ι). Algorithm 1 summarises the training procedure.Algorithm 1: Summary of the training procedure.Input: Budget R; batch size B ∈ 1, R -1 ; ABM M; intervention distribution η; surrogate family M Ψ ; abstraction map family Ω Φ Result: Trained surrogate and abstraction map parameters, ψ * and ϕ * Set D = ∅; for r = 1 to R do Sample ι (r) ∼ η, x (r) ∼ P M ι (r) ; D ← D ∪ (ι (r) , τ (x (r) )) end while not converged do Sample minibatch {(ι (b) , τ (x (b) ))} B b=1uniformly from D; Take gradient step in ϕ, ψ using Equation 9end Definition 4 is closely related to exact transformations: Proposition 1. Let η be an interventional distribution, d be a statistical divergence, and (τ, ω) be a τ -ω transformation between SCMs M and M′ . If τ -ω is 0-approximate (d τ,ω (M, M ′ ) = 0), then we have η-almost- surely τ # (P M ι ) = P M ′ ω(ι) .The proof is in Appendix D.1. In particular, when I is finite and η(ι) > 0 ∀ι ∈ I, then any 0-approximate τ -ω transformation is an exact τ -ω transformation between M and M ′ . This motivates our own choice of loss function: minimising Equation 8 induces ϕ and ψ to produce a surrogate that behaves the same way as the simulator under interventions of interest.Definition 4 employs an expectation over an interventional distribution η. As a result, even when the abstraction error is low, there may still be a large discrepancy between τ # (P M ι ) and P M ′ ω(ι) for some fixed intervention ι ∈ I. Proposition 2 provides an upper bound on the error associated with any intervention sampled from η when d is the KL divergence and the simulator state space is finite: Proposition 2. Let d be the KL divergence and CE ι = E Y∼τ # (P Mι ) -log q ω(ι) (Y) denote the cross-entropy of P M ′ ω(ι) with respect to τ # (P M ι ). Assume dom[X] is finite. Then for all ε > 0,P η d τ # (P M ι ), P M ′ ω(ι) ≥ ε ≤ E ι∼η [CE ι ] ε .The proof is in Appendix D.2. This shows that it is only with low probability that the effects of individual interventions are captured poorly by the surrogate when the surrogate and abstraction map parameters, ψ and ϕ, are found by minimising Equation 8. Here, we outline a case study2 in which we learn interventionally consistent surrogates for the spatial SIRS ABM from Example 1, allowing us to experiment more rapidly with policy interventions while remaining confident that the causal behaviour of the original SIRS ABM is approximately preserved. Further experimental details and results are given in Appendix E. We consider three families of surrogate models with endogenous variables Ĩ0 ∈[0, 1], θt ∈ R 3 ≥0 for t ∈ 1, T , and ỹt ∈ {(a, b, c) | a, b, c ∈ 0, N , a + b + c = N } for t ∈ 0, T, where a, b, c denote, respectively, the number of susceptible, infected, and recovered individuals in the population. The DAGs underlying the SCMs of each of these families are as in Figure 4 , and the three families differ only in the form of the structural equations mapping from Ĩ0 and θ0:t to the ỹt . Throughout, we let q ψ be a Multinomial emission distribution and ψ be trainable parameters of these structural equations.Surrogate family 1 consists of a latent ODE (LODE) built by feeding the classical SIRS ODE's three state variables (which take values in the two-simplex) in as the class probabilities of q ψ . Here, ψ = ∅. Surrogate family 2 consists of a latent ODE-RNN (LODE-RNN), where we run a recurrent network (RNN) with parameters ψ over the output of the SIRS ODE. The RNN outputs the class logits of q ψ . Surrogate family 3 consists of a latent RNN (LRNN) constructed by running an RNN with trainable parameters ψ over the θt , and the output of the RNN at each t ∈ 0, T indexes the class logits of q ψ . Given θ1:T , Ĩ0 , these surrogates enjoy tractable likelihood functions, which factorise as q ψ (ỹ0:T | θ1:T , Ĩ0 ) = q ψ (ỹ 0 | Ĩ0 ) T t=1 q ψ (ỹ t | θ1:t , Ĩ0 ).Interventions & the τ -ω transformation. Denotingι v,a = do (θ 1:T = v, I 0 = a) , (10) ι v,a,t l = do (θ 1:t l -1 = θ t l +6:T = v, θ t l :t l +5 = v ⊙ (0, 1, 1), I 0 = a) ,we define two subsets I = I init ∪ I init, lock of interventions for the ABM: 2.15 2.17I init = {ι v,a | (v, a) ∈ [0, 1] 4 } and I init, lock = {ι v,a,t l | (v, a, t l ) ∈ [0, 1] 4 × 5, 10 }. (11)The first of these is a subset of interventions that fix the initial proportion of infected individuals in the ABM, as well as its parameter values. The second subset of interventions is the set of interventions that fix (a) the initial proportion of infected individuals in the ABM, (b) the values of the ABM's parameters before, during, and beyond a lockdown beginning at time t l ∈ 5, 10 with duration equal to 5 time steps, and (c) the value of t l . Similarly definingEQUATIONι ′ ṽ,ã, tl = do θ1: tl -1 = θt l +6:T = ṽ, θt l : tl +5 = ṽ ⊙ (0, 1, 1), = ṽ, Ĩ0 = ã ,we define I ′ = I ′ init ∪ I ′ init, lockfor the surrogates, where, lettingD = R 3 ≥0 × [0, 1],we haveEQUATIONThe map τ is taken to map: θ t identically to θt for each t ∈ 1, T ; the microstate x t of the ABM at each time step to the ỹt through an aggregation map that counts the number of agents in x t in each of the three states (susceptible, infectious, and recovered); and the initial proportion I 0 of infected agents in the ABM identically to Ĩ0 . Further, for a neural network f ϕ : [0, 1] 3 → R 3 ≥0 , we takeEQUATION)The benefits of training for interventional consistency We use Algorithm 1 to jointly learn the parameters ϕ, ψ of the surrogates and the map ω ϕ described above in two different ways: training the surrogate models with η taken to be a uniform distribution U(I init ) over I init , which entails comparing the behaviour of the surrogate and ABM without lockdowns at different parameters; and training with η instead taken to be a uniform distribution U(I) over I, which entails comparing the behaviour of the surrogate and ABM under different lockdowns, or no lockdowns at all, at different parameters. We indicate the two approaches to training the surrogates with, respectively, bold uppercase O and I. Appendix E details the training procedure and network architectures. We assess the interventional consistency of the surrogates trained in these two ways by computing error metrics on a hold-out test datasetI ′ = {(ι (r ′ ) , y (r ′ ) 0:T )} R ′ r ′ =1 of size R ′ = 1000, generated as ι (r ′ ) ∼ η = U (I) , y (r ′ ) 0:T ∼ τ # P M ι (r ′ ). Specifically, we inspect the average mean squared error (AMSE) between trajectories from the trained surrogates and y (r ′ ) 0:T , and the average negative log-likelihood (ANLL) of this test data under the likelihood of the learned surrogates. Observational consistency is checked on a different hold-out test set O ′ , generated by instead taking η = U(I init ).Table 1 shows these performance metrics evaluated on I ′ and O ′ for all surrogate families and training schemes. We observe that far lower values of the error metrics are obtained by the interventionally, rather than observationally, trained surrogates when assessing interventional consistency. This suggests that training on interventional data can result in more accurate predictions about the effect of interventions in the ABM, and that data drawn from the relevant interventional distributions associated with the ABM should be included during training if the policy-maker intends to perform policy experiments with the surrogate. We also report a minor drop in observational consistency when training with data from the combined intervention set I instead of I init , which can be explained by the overfit of the observationally-trained model on the observational distribution. We also observe that the LODE-RNN -which combines the "mechanistic" SIRS ODE with a flexible RNN -achieves the best interventional and observational consistencies of all surrogates, suggesting that such hybrid approaches to constructing flexible surrogates are promising choices under our proposed method.In Figure 5 , we show an example of a possible negative consequence of failing to train a surrogate on data drawn from the appropriate ABM interventional distributions. In the middle panel, we show the change in the ABM trajectory induced by imposing a lockdown at time t l = 7, while in the left (resp. right) panel we show corresponding trajectories from the interventionally (resp. observationally) trained surrogates under the equivalent intervention learned through our training procedure. While the interventionally trained LODE-RNN correctly predicts that the lockdown effectively impedes the spread of the disease in the ABM, the observationally trained surrogate predicts that the lockdown will temporarily increase infections, before approximately reverting to the behaviour of the model without a lockdown.The use of such a surrogate model in policy experiments when limited computational resources do not permit use of the accurate, high-fidelity ABM of the underlying complex system may therefore have misdirected policy-makers towards suboptimal, and away from effective, interventions. Indeed, while the SIRS ABM predicts that any lockdown is better than no lockdown at all for reducing the number of infections occurring over the simulated time horizon, we see that the observationally trained surrogates often do not predict that no lockdown is the worst intervention in this respect, and in some cases mistakenly predict that no lockdown is the best intervention. For example, the observational LRNN predicts that no lockdown was the best intervention in 1 of 5 training repeats, and was not the worst option in all 5 of 5 training repeats. In contrast, none of the interventionally trained surrogates predict that no lockdown is the best intervention, and only the interventional LODE model predicts that no lockdown is not the worst option (in only 2 out of 5 training repeats). This highlights the potential importance of training surrogate models for interventional consistency when their purpose is to help inform downstream decision-making tasks. Furthermore, this suggests that a possible benchmark criterion in further research on interventional surrogates could be the degree to which different surrogates preserve the ordering of interventions with respect to those downstream tasks of interest.Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: We provide details on the method and training algorithm in Section 4. We describe the experimental setup in Section 5, and provide further details on neural network architectures, training hyperparameters etc. in Appendix E. Finally, we also make code for our experiments publicly available, in order for the reader to be able to reproduce the results.Guidelines:• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 
open access to data and code: 
 Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes] Justification: Open source code is provided on GitHub, along with instructions for how to run the script to reproduce the results.Guidelines:• The answer NA means that paper does not include experiments requiring code. Justification: We have discussed the potential positive societal consequences of this line of work in, e.g., Section 1. While we do not see any reason that our work should be seen as posing any particular risk of threatening harm to society, we recognise that causal simulation modelling in general can be used by malicious actors to improve their own decision making, which could support them to achieve their own malicious goals.Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
related work: 
 Surrogates are often used to expedite simulation-based inference when modelling complex systems [Heppenstall et al., 2021] . Modern approaches rely on established machine learning methods such as random forests [Lamperti et al., 2018 , De Leeuw et al., 2023] , artificial neural networks [Anirudh et al., 2022 , De Leeuw et al., 2023] , support vector machines [ten Broeke et al., 2021] , kriging [Salle and Yıldızoglu, 2014] , and mixture density networks (MDNs) [Platt, 2022] . Our experiments also rely on established machine learning methods to construct surrogates; for example, our LRNN surrogate family resembles that of Platt [2022] , in which MDNs are used to approximate an ABM's transition density. However, in such works, the causal/interventional consistency of the surrogate with respect to the simulator and policy interventions of interest is not considered. In contrast to prior work, our work explicitly details the causal relation between the surrogate and the underlying simulator via causal abstraction, which broadens the scope of surrogate modelling beyond its current use case of expediting calibration to also enable the use of surrogates for policy experimentation.Causal abstraction and exact transformations were introduced by Rubenstein et al. [2017] . Beckers and Halpern [2019]  
conclusion: 
 We propose a rigorous framework for learning interventionally consistent surrogates for complex simulation models, formalised with casual abstraction. This is the first application of causal abstraction to surrogate modelling. Our approach applies to any simulator corresponding to any DAG, and does not require explicit knowledge of the simulator's SCM. Through experiments, we highlight the efficacy of our method against purely observational surrogates that do not learn to match interventional data under equivalent interventions. Using our framework, policy-makers may be able to more rapidly draw insights from complex simulators about the possible effects of interventions -in our experiments, our surrogates simulate approximately three times faster than the original complex simulators -and swiftly prepare effective responses to future crises.Our work naturally suffers limitations. Investigating the sample complexity of abstraction learning would be desirable in future work. Our definition of abstraction error involves an expectation over interventions rather than a maximum as in Beckers et al. [2020] ; this produces a computationally tractable optimisation problem, but introduces the possibility that one or more interventions is captured poorly by the learned abstraction map, even for a low abstraction error. In our experiments, we have assumed surrogate models with tractable and differentiable density functions, permitting us to use a KL divergence within our definition of abstraction error; future work might extend our approach by considering different surrogate families with these properties, such as families based on normalising flows [Tabak and Vanden-Eijnden, 2010] , or alternative divergences that relax the requirement for tractable densities, such as maximum mean discrepancies [Gretton et al., 2012] . Finally, our method does not directly exploit knowledge of the simulators' causal graphs to accelerate abstraction learning. It is possible that exploiting access to the base SCM/DAG may expedite abstraction by allowing us to focus on minimal intervention sets [Aglietti et al., 2020, Lee and Bareinboim, 2018] , or leverage the identifiability of interventional distributions to reduce the number of simulations required from the base model [Lattimore et al., 2016 , Bilodeau et al., 2022] . However, it is unclear whether or not applying the do-calculus on large causal graphs is more efficient than simulating interventions directly. The "black-box" nature of our approach may be beneficial for this reason, and since it does not require the modeller to explicitly write their simulator as an SCM, making it generically applicable.  
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [Yes] Justification: We have discussed the potential positive impact of our work in, e.g., Section 1, namely that our methods may assist policymakers to more rapidly identify policy interventions that save lives and mitigate economic costs during pandemics and other crises. As discussed in point 9. of this checklist, we do however recognise that simulation modelling and machine learning in general can be misused by malevolent actors to support their own decision-making.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.• The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 11. Safeguards Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA] Justification: We do not see that our method for learning surrogate models poses a particularly high risk for misuse. Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [NA] Justification: We do not use existing assets. Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. 13. New Assets Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [Yes] Justification: Code for reproducing the results is provided on GitHub. The README contains instructions for how to reproduce the results, while the LICENCE file specifies the licence attached to the code. Guidelines:• The answer NA means that the paper does not release new assets.• Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. • The paper should discuss whether and how consent was obtained from people whose asset is used. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: Definition 1 implies the standard assumptions of (i) acyclicity of the DAG G M and (ii) causal sufficiency, meaning that there are no unobserved confounders [Pearl, 2009 , Peters et al., 2017] . These two assumptions entail that our SCMs are Markovian.We also assume faithfulness, guaranteeing that independencies in the data are captured in the graphical model Spirtes et al. [2000] .As In addition, Beckers and Halpern [2019] assume the intervention map ω can be implicitly defined by the map τ , and require the abstraction map τ to be consistent. That is, the image of I under the intervention map induced by τ must equal I ′ . Since we do not couple the maps τ and ω we enforce no such condition. Additionally, Beckers et al. [2020] enforce surjectivity of τ . Since this makes no practical difference in a surrogate's use in downstream tasks, we dispense with this assumption.Alternative notions of abstraction error have been introduced by Zennaro et al.[2023b], building upon the notion of exact transformations introduced by Rischel [2020]. We conjecture that an analogous version of our framework may be developed for this setting, wherein the aggregation function over the intervention set is again chosen to be an expectation over an interventional distribution η instead of a maximum, and we leave this as a direction for future work.Surrogate modelling of complex simulators is closely related to the problem of simulation-based inference. Inference involves tuning model parameters so that data generated by the simulator matches that generated by the real world system being modelled. Analogously, surrogate modelling consists of tuning surrogate parameters so that data generated by the surrogate matches data generated by the corresponding simulator. Hence, methods for calibration can naturally be applied to learn surrogates. Several calibration techniques and metrics have been proposed in the literature, including the method of simulated moments [Fabretti, 2013, Gilli and Winker, 2003 ] and minimum simulation distance [Grazzini and Richiardi, 2015] . We refer the reader to Dyer et al. [2024] for a thorough survey. Unlike our framework, surrogates trained for the purpose of parameter estimation do not typically account for interventional consistency explicitly.More generally, our framework bears similarities to latent space modelling of Markov decision processes (MDPs) [Gelada et al., 2019] , wherein one attempts to learn a smaller latent MDP from a target MDP, whose size precludes its use in downstream tasks. For downstream tasks such as formal verification of policies, Delgrange et al. [2022] employs the bisimulation metric to measure the consistency of their latent MDPs with respect to the target. Abstraction error plays an analogous role in our framework, where the original MDP corresponds to the simulator, and the latent MDP the surrogate. Likewise, the surrogates we propose in Section 5 are implicitly connected to the scientific modelling framework of Rackauckas et al. [2021] , who embed prior information regarding system dynamics into systems of universal differential equations represented by neural architectures such as neural ODEs. We embed the underlying dynamics of the classical SIRS ODE into several surrogates in an attempt to learn better causal abstractions. Our work also bears some similarities to, yet differs substantially in several key ways from, Kekić et al. [2023] , who also use an abstraction error to learn reduced causal models from larger SCMs. While their approach focuses on a single target variable at a fixed time horizon, assumes Gaussian noise and linear structural functions, and focuses on explainability of outcomes, we track multiple interdependent variables over the entire time horizon with a focus on accurate simulation from interventional distributions. Our approach is therefore more tailored to large-scale and realistic nonlinear simulators. In contrast, the method presented in Kekić et al. [2023] becomes impractical for large-scale models.We provide here some practical examples, beyond the two case studies we present, that illustrate how the τ map may be chosen for large-scale simulators, as a guideline for practitioners. We consider three exaples from the literature on policy modelling below:1. Consider the model of forced migration in Ghorbani et al. [2024] . Variables of interest to these modellers are the total number of displaced people by location over time by age, gender, and other demographic characteristics. τ would therefore be defined by counting the number of agents in each of these states at each location, i.e. τ l,d (x t ) =a∈A I[agent a has demographic features d and is in location l at time t] where l indexes locations, d are demographic features, x is the state of the simulation at time t, A is the set of all agents, and I is the indicator function.P η d τ # (P Mι ) ∥ P M ψ ω ϕ (ι) ≥ ϵ ≤ d τ,ω ϕ M, M ψ ϵ .Since we have a finite domain, the likelihood functions associated with (a) the pushforward measure of the ABM under τ and (b) the surrogate macromodel can be written as probability mass functions, whose logarithms are non-positive. Since we have assumed P M ψ ω ϕ (ι)≪ τ # (P Mι ), we have that 0 ≤ -log q ψ ω ϕ (ι) (Y) < ∞ for any Y ∼ τ # (P Mι ), and thereforeEQUATIONWe also have thatEQUATION) ODE Solve θt+1 z t z t+1 ũt+1 MN ỹt+1Figure 6 : A schematic representation of the LODE surrogate family for a single time step. First, the output of the SIRS ODE for the next time step, z t+1 , is computed via ODESolve. Then, z t+1 serves as the logits for a multinomial distribution from which ỹt is sampled. This sampling procedure is denoted by MN in the diagram. The exogenous variables required to reparameterise the multinomial distribution during sampling are denoted by ũt .where H τ # (P Mι ) is the entropy of the probability mass function associated with τ # (P Mι ), and thatEQUATION)We write the upper bound above in terms of the cross-entropy, since this can be estimated with finite samples, whereas the full KL-divergence cannot be estimated in general due to the complexity of evaluating the density associated with τ # (P M ι ) for an arbitrary ABMs. HenceEQUATION)As described in the main text, the three surrogate families we consider have SCMs whose corresponding DAGs can be drawn as in Figure 4 . In this section, we fully specify the corresponding SCM for each surrogate. Furthermore, for each surrogate, we provide details on the procedure used to train the parameters ψ and ϕ, which respectively describe the structural equations of each SCM and their corresponding intervention map ω.To construct a set M of probabilistic SCMs, we define a latent neural ordinary differential equation (LNODE) based on the classical SIRS ODE system. The SIRS ODE system takes the formEQUATIONwhere θt = (α t , βt , γt ) ∈ R 3 ≥0 are the ODE parameters and z t = ( St , Ĩt , Rt ) ∈ S ∀t ∈ [0, T ] is the ODE state, where S is the two-simplex. Note that z t represents the proportion of susceptible, infected and recovered individuals in the population according the SIRS ODE. Whilst the parameters θt may change over time -which will permit the experimenter to intervene on the values of the parameters at different time steps -we assume the simplest case of assigning the same vector ṽ ∈ R 3 ≥0 to all θt when no interventions are applied:EQUATIONIn other words, Equation ( 21) describes the structural equation fθ t for θt . Practically speaking, the choice of ṽ is inconsequential, as we can model any change to θt as an intervention. Given θt , the ODE state z t evolves according to the following rule:EQUATIONwhere ODESolve denotes numerical integration of System 20 between times t -1 and t. In our experiments, we compute this using a Euler scheme with step size ∆t = 1. The initial state of the ODE Solveθt+1 z t z t+1 h t RNN h t+1 FF o t+1 ũt+1 MN ỹt+1Figure 7 : A schematic representation of the LODE-RNN surrogate family for a single time step. First, the output of the SIRS ODE for the next time step, z t+1 , is computed via ODESolve. Then, z t+1 is passed through to the hidden state of a recurrent neural network (denoted by RNN in the diagram) that updates its hidden state from h t to h t+1 . The updated hidden state is passed to a feedforward neural network (denoted by FF in the diagram), which computes the logits o t+1 for a multinomial distribution from which ỹt+1 is sampled.ODE is taken to be z 0 = (1 -Ĩ0 , Ĩ0 , 0). One may change the initial state z 0 through interventions on Ĩ0 , which is modelled as an endogenous variable.Given z t , we draw the endogenous variables ỹt from a multinomial distribution whose class probabilities are given by z t . Whilst z t represents the percentage of susceptible, infected, and recovered individuals predicted by the SIR ODE, ỹt represents the actual counts observed by the experimenter. We write fỹ t ( Ĩ0 , θ1:t , ũt ) to denote the structural function associated with ỹt , where the dependence on Ĩ0 and θt ′ for t ′ ≤ t is mediated by the trajectory followed by the z t ′ for t ′ ≤ t, and ũt are the exogenous random variables required to reparameterise the multinomial sampling procedure on each time step.Note that ψ = ∅ for this family of surrogates, and hence M is a singleton. For the function f ϕ comprising the intervention map ω ϕ , we take a feedforward network with layer sizes 3, 32, 64, 64, 64, 32, 3. A ReLU activation is applied after each hidden layer, and a sigmoid activation is applied to the final output layer. The sigmoid activation function ensures that the predicted intervention vector f ϕ (v) on the parameters of the LODE has all of its components in the range [0, 1], which is suitable when forward simulating the ODE with an Euler scheme with ∆t = 1. This feedforward network consists of 12,739 trainable parameters.This surrogate family closely mimics the LODE family described above, and differs only in that the class logits of the multinomial distributions are instead indexed by the output of a feedforward network -with layer sizes 32, 32, 64, 32, 16, 3 , where all hidden layers are followed by a ReLU activation function -which maps from the hidden state h t ∈ R 32 of a GRU recurrent network that is passed over the trajectory z 0:T generated from the SIRS ODE (forward simulated as described above).The combined action of the ODE solver, the GRU-feedforward networks, and the reparameterisation of sampling from the multinomial distributions, define the structural equations fỹ t : ( Ĩ0 , θ1:t , ũt ) → ỹt for each t ∈ 1, T .For this model, ψ is the collection of trainable parameters comprising these GRU and feedforward networks. For f ϕ , we use a feedforward network with layer sizes 3, 32, 64, 32, 3, where a ReLU activation is applied after all hidden layers and a sigmoid activation is applied after the final layer. Thus, the total number of trainable parameters from ψ and ϕ combined is 13,798.This surrogate family makes no use of the SIRS ODE model. Instead, the logits of the multinomial distributions for t ∈ 1, T are indexed by the outputs (o 1 , . . . , o T ), o t ∈ R 3 of a feedforward network -with layer sizes 32, 32, 64, 32, 16, 3 , and where all hidden layers are followed by a ReLU activation function -that maps from the hidden state h t ∈ R 32 of a GRU recurrent network which is passed over the sequence θ1:T . The initial hidden state is chosen to be h 0 = (1 -Ĩ0 , Ĩ0 , 0), where 0 is a vector of 30 zeros. We also take o 0 = (log(1 -Ĩ0 ), log( Ĩ0 ), -∞) which indexes the logits of the multinomial distribution at time t = 0. Once again, we may write the structural equations fỹ t for the ỹt in terms of Ĩ0 , θ1:t , and the exogenous random variables ũt required to reparameterise the sampling procedure from the multinomial distribution.h t h t+1 FF o t+1 MN ũt+1 ỹt+1Figure 8 : A schematic representation of the LRNN surrogate family for a single time step. First, the parameters θ t+1 are passed to a recurrent neural network (denoted by RNN in the diagram) that updates its hidden state. The updated hidden state is passed to a feedforward neural network (denoted by FF in the diagram), which computes the logits o t+1 for a multinomial distribution from which ỹt+1 is sampled.Since we use exactly the same networks in this surrogate family as in the LODE-RNN family, the total number of trainable parameters from ψ and ϕ combined is also 13,798.Having intervened on the Ĩ0 and θt with known values, the class probabilities for each multinomial distribution is completely determined given the deterministic dynamics within the structural equations mapping to the ỹt .Takingdom[I 0 ] = J M = [0, 1], dom[X 0:T ] = X T +1 with X = {0, 1, 2} N , and dom[Θ 1:T ] = P T M with P = [0, 1] 3 , we define τ : J M × X T +1 × P T M → J M ′ × Y T +1 × P T M ′which operates componentwise asEQUATIONwhereEQUATIONEQUATIONIn the above, id is the identity map, and τ x acts by counting the total number of susceptible, infected, and recovered individuals in the ABM at each time step.All models were trained on CPU on a 2022 MacBook Pro, operating on macOS Ventura 13.2.1.Training one surrogate model on this machine took on average approximately 20 minutes, amounting to approximately 600 minutes in total to produce the results reported in Table 1 . Initial attempts at experiments while the code was still in development contribute approximately 200 additional minutes. Software dependencies are specified in the GitHub repository containing the code for this paper, which will be made public upon acceptance.We assume periodic boundary conditions in both spatial dimensions for the ABM presented in Example 1, which is used in all of our experiments.As suggested by Figures 1, 4 , and 6-8, the parameters θ and θ are fed into the models at each time step.For the LODE and LODE-RNN surrogate families, we forward simulate the SIRS ODE with an Euler scheme with step size ∆t = 1.For all surrogates, the neural networks comprising the ω ϕ map and structural equations parameterised by ψ were trained with a learning rate of 10 -2 for a maximum number of 1000 epochs, batch size B = 50, and with the Adam optimiser [Kingma and Ba, 2014] . A total number of R = 1000 training samples was generated from the ABM for each of the observational and interventional training sets; these were each split 5 times into different training and validation sets of sizes 800 and 200, respectively, with a new surrogate model trained from scratch on each of these splits. We apply an early stopping criterion in which training is ceased if the validation error does not decrease for 20 consecutive epochs.In this case study, we consider a different policy scenario: reintroducing a species into an ecology, and simulating the ensuing population dynamics. Specifically, we adapt slightly a model from Wilensky and Reisman [2006] : we model an environment initially consisting of grass, sheep, and wolves, in which grass grows and is eaten by sheep, sheep eat grass and reproduce and are eaten by wolves, and wolves eat sheep and reproduce. The intervention we consider entails reintroducing a third animal species -bears, which eat both sheep and wolves, and also reproduce -whose population is originally zero but is made non-zero at some intervention time t. We imagine that t is the variable the policymaker wants to optimise here.We simulate the interactions between these four species in a spatial model, in which members of each animal species move around the grid and interact with the other species. We are then interested in understanding how the reintroduction of the bears affects the overall population dynamics, i.e., the counts of each animal in each species, along with the quantity of grass over time. As in the epidemic case study, we consider the problem of learning interventionally consistent surrogates for this complex spatiotemporal simulator, and once again examine three possible approaches for constructing surrogate families:1. a family of deterministic mechanistic models based on a discrete-time Lotka-Volterra model of population dynamics [see, e.g., Sabo, 2005] , where (analogously to the LODE surrogate family discussed in the epidemic case study) the underlying deterministic dynamics of the population dynamics model index a probability distribution at each time step (in this case, a Binomial distribution for each of the 4 species); 2. an LRNN family, exactly mirroring the LRNN family considered in the epidemic case study presented already; 3. and a third family considers a hybrid approach, where (as in the LODE-RNN family considered in the epidemic case study) we pass a recurrent network over the underlying Lotka-Volterratype population dynamics model first before taking the output of the recurrent network to index the Binomial distributions for each of the four species.A table for the results of this additional case study is shown in Table 2 , where we see that the results are qualitatively very similar to the epidemic case study already presented: we see that training surrogates using our framework yields significant improvements in the surrogates' interventional consistency over observationally trained baselines, and that interventionally trained surrogates only see a minor decrease in performance on observational data compared to the drop in performance the observational surrogates see on interventional data.Justification: These are discussed in the Conclusion (Section 7).Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.Code for reproducing the experimental results is available at https://github.com/joelnmdyer/ neurips_ics4csm. 


Decision: This paper introduces a robust framework for learning surrogate models that maintain consistency under interventions, specifically designed for large-scale complex simulations. After a thorough review of the content, I recommend accepting the paper for the following reasons. The key strength of this work lies in its innovative approach to ensuring that surrogate models behave consistently under various interventions, a critical requirement for accurate and reliable simulation in policy-making and other high-stakes environments. The authors build upon established causal abstraction theories and provide strong theoretical foundations that justify their method. Their approach is particularly noteworthy for its ability to scale to large, complex systems, which are common in real-world applications but challenging for existing methods. The empirical validation, conducted on an epidemiological model, effectively demonstrates the method’s practical utility. The surrogate models trained with the proposed framework showed significant improvements in consistency under intervention, reducing the risk of erroneous policy decisions based on simulation results. The authors also address potential concerns about the method’s applicability to other domains by discussing extensions and providing a new case study in the rebuttal. Importantly, the criticism raised by Reviewer x5Gt regarding similarities with prior work is not a valid basis for rejection. The authors have convincingly demonstrated that their approach is substantially different from and more general than the work in question, particularly in its application to more complex, multi-variable systems and its scalability. This distinction highlights the novelty and impact of the paper, making it a significant contribution to the field. Given the strong theoretical contributions, empirical validation, and the clear distinction from contemporaneous work, this paper is a valuable addition to the literature and should be accepted.

Reviews: review: paper_summary: paper_summary: The paper introduces a framework for learning surrogate models for complex simulations that preserve simulation behavior under changes in the structural parameters of the underlying model (i.e. the intervention) using causal inference. The framework is tested on epidemiological agent-based models of disease spread, against a set of ablative baselines.
strength_weakness: strength_weakness: W1: While the SIRS model provides a good initial test case, the empirical evaluation is limited to a single domain. It would strengthen the paper to demonstrate the method's applicability to other types of complex simulations, such as economic or social systems models.

W2: The paper does not provide a thorough comparison to existing surrogate modeling techniques beyond a basic observational training baseline. It would be extremely valuable to see how the proposed method compares to state-of-the-art surrogate modeling approaches, even if they don't explicitly consider interventional consistency.

W3: The scalability of the proposed method to very large and complex simulations is not thoroughly addressed. It's unclear how well the approach would work for simulations with hundreds or thousands of variables and complex interdependencies.

W4: The paper lacks a discussion of how to choose appropriate interventions for training the surrogate. In practice, the space of possible interventions may be very large, and it's not clear how to select a representative set for training and testing.
questions: questions: Q1: How sensitive is the method to the choice of the interventional distribution $\eta$? How would you recommend practitioners choose this distribution in real-world applications? If you were to read the paper afresh, what would you need to have in the manuscript to be able to expand its experimental results so as to form a consistent set of benchmarks for the community?

Q2: Have you explored the performance of the method under model misspecification, i.e. where the surrogate family does not include a model that can perfectly capture the behavior of the original simulator?

Q3: The paper mentions that the method does not require explicit knowledge of the simulator's SCM. How does this compare to methods that do utilize such knowledge, and are there cases where having this knowledge would be beneficial (or where it's a necessity and/or the proposed method critically fails)?

Q4: How does the computational cost of training the surrogate compare to running the original simulator? Is there a break-even point in terms of the number of interventions one needs to evaluate for the surrogate to be worthwhile?
limitations: limitations: L1: The current framework assumes that the map $\tau$, which defines the aggregate quantities of interest, is pre-specified. In practice, determining the right level of abstraction and which quantities to preserve may be challenging. The paper would benefit from a discussion of how to choose appropriate τ maps.
paper_summary: paper_summary: The paper proposes a novel framework for learning surrogate models of expensive simulators by formulating them as structural causal models. Specifically, the authors focus on learning interventionally consistent surrogate models for large-scale complex simulation models. The authors' claims are supported by theoretical and empirical results.
strength_weakness: strength_weakness: a. Empirical tests seem to be limited in assessing the framework's true power. Tests on different, perhaps more complex, scenarios would help practitioners to better understand the benefits of the proposed framework and use it.

b.  One motivation for the proposed framework is to provide policy-makers with "fast" and high-fidelity surrogates to test the effect of different interventions. However, the empirical results do not clearly indicate what the advantage of surrogates compared to simulators is in terms of speed of simulations.
questions: questions: See Weaknesses
limitations: limitations: Limitations of the work are reported in section 7
paper_summary: paper_summary: This work presents a method for learning surrogate models of simulations.
Both the simulation and the surrogate model are treated as causal models, where the surrogate is an element of a parametric family of causal models.
The map from simulation variables to surrogate variables is assumed to be known.
For training the surrogate, interventional consistency is used: interventions in the simulation should be well-represented by the surrogate.
That is, a map from simulation interventions to surrogate interventions has to be learned.
As a training loss, an abstraction error is used, which measures the distance between the pushforward of intervened simulation variables and intervened surrogate variables.
The theory section shows that zero approximation error leads to an exact transformation and that approximation errors for a given intervention are bounded.
The approach is demonstrated on an agent-based simulation of an epidemic model (SIRS), with interventions corresponding to different spreading rate settings (some represent lock-downs).
strength_weakness: strength_weakness: While I very much enjoyed reading the paper, it is unclear to me what its contribution is with respect to previous literature. The main elements (e.g. treating simulations as causal models and learning a simpler surrogate, making the connection to causal abstractions and using some form of approximation error for training) have been introduced before in [1] with only minor differences as far as I can tell.

References:

[1] Kekić et al. "Targeted Reduction of Causal Models." UAI (2024)
questions: questions: - L96 "we restrict our attention to hard interventions": It looks like you don't use this fact later on in the theory. Could you relax this requirement? In fact, you could argue that the interventions applied in the case study look more like soft interventions. You intervene on $\theta$, which really just encodes the parameters of the causal mechanisms responsible for propagating infections and other state variables through time. So theta looks more like something that encodes some mechanisms, rather than a state variable.
- L216 "we assume that the base model M is implicitly represented by a simulation model of a complex socio-technical system": Is this just a helpful picture to have in mind for the reader, or is there some technical assumption that you make? It doesn't seem to me that this is used later on. Maybe I'm confused by "socio-technical system": for what type of simulations would the method not work?
- Fig. 4: Why are there edges from $\tilde{\theta}$ to $\tilde{y}_2$ and to $\tilde{y}_3$? I thought $\tilde{\theta}_t$ encodes the transition parameters from one time step to the next. Similarly, for $\tilde{I}_0$: wouldn't it only affect the first time step?
limitations: limitations: The method has some severe limitations in how easily it can be applied to real-world simulations, as it does not specify how to tackle the following:

- How to make the distribution of the surrogate family tractable and differentiable.
- How to select the surrogate family.
- How to find the variable map between the simulation and the surrogate.

These elements are assumed to be given, but I think they are the main difficulty when modelling simulations. I understand that modellers need to provide some level of domain expertise and that makes every application somewhat different. But the paper doesn't give much guidance on how the difficult points above can be addressed.

While Prop. 1 and 2 are reassuring, they don't give the modeller many insights as to how the approach behaves. For example, how do you need to choose the set of interventions in order to get a good surrogate? That is touched on in the case study, but to make the method more reliable in practice it would be great to have some more general theoretical analysis. Another open question is: What happens when you misspecify the surrogate by choosing a causal structure that doesn't match that of the simulation?
paper_summary: paper_summary: In their use by practitioners, complex simulators may be run under intervention scenarii. To reduce the overall cost of running many scenarii, relying on cheaper to compute surrogate models is a standard practice. The authors thus propose surrogates that can additionally learn these interventions effect, relying on a causal inference representation. Specifically, it divides the training into a parametric surrogate family and abstraction map parameters that are learned with the Kullback-Leibler divergence. An epidemiology example is provided to show the interest of learning this decomposition on surrogate consistency.
strength_weakness: strength_weakness: - the discussion of the results could be improved
- the test case is perhaps a bit simple
questions: questions: - the naive way for surrogate modeling in this context (say with neural networks or random forests) would be to also give as inputs the intervention parameters corresponding to the observation. It is not completely clear to me if this is the case for the RNN in the example.
- More analysis on the performance of interventionally trained data on observational data would be beneficial.
- perhaps some discussion on surrogates with causal inference would be relevant, see e.g., 
Witty, S., Takatsu, K., Jensen, D., & Mansinghka, V. (2020, November). Causal inference using Gaussian processes with structured latent confounders. In International Conference on Machine Learning (pp. 10313-10323). PMLR or Toth, C., Lorch, L., Knoll, C., Krause, A., Pernkopf, F., Peharz, R., & Von Kügelgen, J. (2022). Active bayesian causal inference. Advances in Neural Information Processing Systems, 35, 16261-16275.
- A related topic is the use of monotonicity information, e.g., on intervention effect, see e.g., Riihimäki, J., & Vehtari, A. (2010, March). Gaussian processes with monotonicity information. In Proceedings of the thirteenth international conference on artificial intelligence and statistics (pp. 645-652). JMLR Workshop and Conference Proceedings.
limitations: limitations: NA

Comments: title: (2/2)
(1/2)
(2/2)
(1/2)
References in Rebuttal
comment: Thank you again for your feedback – we hope we have been able to address your remaining concern about the limited role that domain knowledge plays in defining the variables of interest to the modeller. If you believe our improvements warrant an increase in your initial score, we would greatly appreciate seeing this be reflected in your updated score prior to the deadline in ~20 mins. Thank you again!
Thank you once again for your feedback – we hope we have been able to address your questions in our responses. If you believe our improvements warrant an increase in your score, we would be very grateful to see this reflected in your updated score ahead of the deadline in ~25 mins.
Thank you for your feedback and for your considered assessment of our work. If you believe our improvements warrant an increase in your initial score, we would greatly appreciate if this could be reflected in your updated score prior to the deadline in ~25 mins.
- Reviewer `x5Gt` continues to call our solution "handcrafted", which _completely ignores_ the fact that we learn two elements that are crucial for the accuracy of the high-level simulator:

    **a.** the omega map (which is crucial for correctly mapping interventions between models) and 

    **b.** the mechanisms governing the relationships between the high-level variables (which is crucial for simulating accurately from the high-level interventional distribution). 

    Therefore, Reviewer `x5Gt`'s claim that our solution is handcrafted is false, and appears to be a deliberate attempt to misrepresent our contribution.
- Reviewer `x5Gt` claims that our contributions are not novel relative to [R3.1], yet we have demonstrated experimentally that the algorithm proposed by [R3.1] does not scale to our settings. In contrast, our own approach does. This drastic improvement cannot be achieved without novelty (else [R3.1]'s approach would already achieve this improvement). [R3.1] themselves highlight in their Limitations and Future Work section that "future work should explore more flexible approaches" than linear Gaussian abstractions, which further supports the novelty of our approach.
- Reviewer `x5Gt` claims that our work is of limited use to the ML community, yet we run our learned surrogates on case studies that are _significantly larger_ in scale and complexity than those that [R3.1]'s approach can handle.
- Reviewer `x5Gt`'s point about inductive biases actually highlights a strength of our method: we rely on _generic neural networks_ to learn the crucial components of our solution, while [R3.1] adopts a specific inductive bias of _linear_ Gaussian methods (even simpler than the tailored _polynomial_ method that Reviewer `x5Gt` offers themselves).
- Reviewer `x5Gt`'s claim that our work is of "limited impact" is _incompatible with Reviewer `x5Gt`'s own assessment_ that our work "bridges the gap between the causality ivory tower and useful applications."
- We have extensively covered the "problem" of determining $\tau$ in our discussion. Please see, for example, responses (1/2) and (2/2) to Reviewer `fUGP`. The domain knowledge we assume is only that the modeller knows what they are trying to model (which is always the case when a modeller builds a model), and is therefore minimal.
- The additional case study is not "promised", but is expressly included in the Global Rebuttal and accompanying PDF file.
Thank you for the detailed replies to my comments and for proposing a newer test case. I think there is room for this work that relies partly on domain knowledge, but there is an effort to include the above responses.
Thank you for providing additional clarifications and arguments. To me, it seems we are converging to a state where a big chunk of the arguments are repeated, so I'll try to be brief in my answer.

## On the comparison to [R3.1]

Of course, a handcrafted solution to a specific problem scales better (in terms of compute and number of parameters) and achieves better losses than a general-purpose method. 
If I want to fit a function and I know it's a polynomial, I can achieve better scalability and results by using a polynomial inductive bias rather than, say, a neural network.

I think the surrogate learned for the epidemiological case study is appropriate and summarizes the original system well.
But besides the submission claiming false novelty, my main criticism is that it doesn't provide a generally applicable method.

## Summary

While I appreciate the effort of adding detailed answers in the discussion, I'm afraid the main criticisms haven't been resolved:

- The paper makes inaccurate claims of novelty, since a large part of the reported contributions have been covered by prior work.
- The paper presents a case study for learning surrogates for one application (or two if you take the example promised during the rebuttal). However, it relies on domain knowledge to define crucial elements of the surrogate map and model. Therefore, it is of limited use to the wider ML/simulation community.

Therefore, I will keep the score of 2 based on "limited impact" (as in the definition of the score).
Thank you all for your feedback during the review and discussion period. If there are any further points that you would like to discuss before the discussion period closes, please do let us know and we will be happy to respond. 

We are particularly cognizant of the extreme discrepancy between the positive scores and feedback we have received from all reviewers (7 from `fR4E`, 6s from `fUGP` and `3hmr`, positive comments from all reviewers including `x5Gt`) and the single outlying "Strong Reject" score from Reviewer `x5Gt`, whose concern is the relation of our work to a paper appearing at UAI '24 approx. 4 weeks ago (herein [R3.1]). 

We have already used the rebuttal and discussion to highlight the many substantive differences between our work and [R3.1]'s. We have shown that [R3.1] differs substantially in focus from our work, and is entirely unsuitable for the settings we consider, given that it scales extremely poorly to the large-scale complex simulators of interest to us. We have also carefully consulted [R3.1]'s paper and used [R3.1]'s accompanying code to run computational experiments that verify these claims. (Please see the discussion below for details.) 

While a "Strong Reject" is, therefore, inappropriately low and incompatible with Reviewer `x5Gt`'s review and the ensuing discussion*, we would in particular like to invite any further comments from the reviewers on any concerns they may have in relation to [R3.1], in order that we may address them. And if you have no further concerns about our work, please do champion our paper!

Thank you once again for your helpful feedback.

*A "Strong Reject" is reserved for papers "with major technical flaws, and/or poor evaluation, limited impact, poor reproducibility and mostly unaddressed ethical considerations". In contrast, all reviews give "Soundness: 3: good" for our work, rather than claiming major technical flaws; "poor evaluation" is incompatible with the fact that we have provided two case studies, each studying 3 approaches to learning interventionally consistent surrogates, and conducted an experimental comparison against [R3.1]'s approach; "limited impact" is incompatible with the fact that **our work currently provides the only viable (practical, scalable, and interpretable) method to learn interventionally consistent surrogates for complex simulators** of the kind used to inform decision-making in complex systems; "poor reproducibility" is not raised by any reviewer; and no reviewers raise ethical concerns.
As further evidence of the inapplicability of [R3.1]'s work to the settings we consider (and, therefore, as further evidence of the significance of our contribution), we have run an experimental comparison against [R3.1]'s approach. We used their code on GitHub (released in June, a month after the NeurIPS submission deadline) and followed the setup for their "double well" experiment. Details are below. 

**Key takeaways**:

- To run [R3.1]'s approach, we were forced to run reduced experiments on a far smaller scale (`N=64` agents, `T=20` time steps) compared to what we are able to do with our own methods, because of the extremely poor scalability of [R3.1]'s approach (which we highlighted in our previous response).
- Even in this extremely small-scale example, [R3.1]'s approach demands ~150k trainable parameters, and requires ~6 hours of training to learn high-level models for all `3T` `S_t`, `I_t`, and `R_t` variables. On this basis alone, it does not compete with our methods: our (off-the-shelf) `LRNN` surrogate family requires ~2 mins to train a single high-level surrogate simulator for _all_ `S_t`, `I_t`, and `R_t` variables.
- Even in this extremely small-scale example, [R3.1]'s approach achieves an MSE loss of ~8.7 on its ability to recover the `S_t` alone. In contrast, our `LRNN` surrogate family achieves an MSE loss of ~0.6 _across all_ `S_t`, `I_t`, and `R_t`. The `LRNN` achieves this by learning (a) how to map interventions from the base model to the abstract model (through a learnable $\omega_{\phi}$) and (b) the mechanisms governing the evolution of the high-level variables (i.e. through learnable surrogate parameters $\psi$).

For these reasons, regardless of whether [R3.1] constitutes prior or contemporaneous work, our contribution is significant and impactful: currently, our approach provides the only viable (practical, scalable, and interpretable) method to learn interventionally consistent surrogates for complex simulators of the kind that are used to inform decision-making in complex systems. 

In light of this and of the discussion above, the reviewer's score and position – which is based on their "primary concerns regarding the novelty and applicability of the proposed method, especially in light of prior work" – is not justified.

### Details

In [R3.1]'s double well experiment, the motion of a ball is simulated according to an equation of motion corresponding to a double well potential. The ball's velocity is shifted at different time steps, while the ball's position is unshifted throughout (see App. F.2 of their paper). The authors of [R3.1] learn $\tau$ and $\omega$ maps from the ball's positions and velocities over all times $t=1,...,T-1$ to the final position of the ball at end time $T$ (see Sec. 5.2 of their paper). Following this example, learning a targeted causal reduction to a variable at time $t$ would involve learning $\tau$ and $\omega$ maps from all variables in the low-level graph corresponding to times preceding $t$. Therefore, applying [R3.1]'s approach in our setting involves learning $\tau$ and $\omega$ maps from all agent states (the $N$-dimensional $x_t$ in our notation) and parameters ($\theta_t$ in our notation) for $t=1,...,t'-1$ for any target variable occurring at time $t'$. We train both [R3.1]'s approach and our `LRNN` surrogate for 40 epochs ([R3.1]'s approach is too computationally demanding to train for longer) and train for 5 different repeats.
## More detailed response

1) Using human knowledge & expertise is ubiquitous in areas of interest to the NeurIPS community. For instance, universal differential equations [1] require domain experts to specify an ODE roughly capturing the physical laws underlying a system. Further, many modern neural architectures rely on physical symmetries found through human domain expertise to improve efficiency (e.g.: permutation symmetries in graphs [2], equivariances in NNs for point cloud data [3], time reparameterisation invariance in NNs for time series [4]). Other active areas of research that incorporate strong biases from human knowledge include physics-informed NNs [5] & human-in-the-loop RL [6].

   **Our use of domain knowledge is minimal** in comparison to these areas. We only require the modeller to supply the variables of interest to them so that $\tau$ is defined. (The modeller generally _will_ know what they are interested in using the model for: if they have sufficient domain knowledge to build a model, they will know what they want to achieve with the model, i.e. how to define $\tau$.) Our minimal reliance on human input is exemplified by our `LRNN` surrogate, which is an off-the-shelf/general purpose network that reproduces the complex simulators' behaviours of interest by taking in interventions through the learned $\omega$ map. 
   
   We will however provide in our revision several detailed practical examples (beyond the two case studies we present) illustrating how $\tau$ may be chosen for large-scale simulators, as a guideline for practitioners. Some examples we will expand on in the revision:
- Consider the model of forced migration in [7]. Variables of interest to these modellers are the total number of displaced people by location over time by age, gender, and other demographic characteristics. $\tau$ would therefore be defined by counting the number of agents in each of these states at each location, i.e. $\tau_{l,d}(x_t) = \sum_{a\in A} \mathbb{I}[\text{agent }a\text{ has demographic features }d\text{ and is in location }l\text{ at time }t]$ where $l$ labels locations, $d$ are demographic features, $x_t$ is the state of the simulation at time $t$, $A$ is the set of all agents, & $\mathbb{I}$ is the indicator function.
- Consider the model of flood risk mitigation behaviours in [8]. The modellers want to model what precautions households take to protect themselves from floods in high-flood-risk areas under different policy interventions. Households can: do nothing; purchase insurance; purchase property-level protection; or purchase property-level protection & insurance (see Fig. 3). Here, $\tau$ would count the number of households taking such actions in this case (as in the example above).
- Consider the UK housing market model in [9]. Tables 2-6 define macroeconomic indicators such as inflation rates, unemployment rates & real interest rates that the modellers care about. Here, $\tau$ would be defined by standard macroeconomic formulae for these quantities.

2) Various guidelines on the **responsible use of ML** (e.g. those highlighted in our TL;DR) recommend that ML **should always be applied in conjunction with domain experts** to adhere to principles governing the responsible use of ML. This is especially true when it comes to high-stakes decision-making in complex systems (the setting we consider). In this way, rather than being a limitation of our approach, our assumption that $\tau$ is informed by a human expert provides a useful & important way to integrate human knowledge into the surrogate by allowing the decision-maker to fix the properties they care about for their decision-making – all while still allowing for flexibility through the learnable $\phi$ and $\psi$ parameters of the $\omega$ map & surrogate itself, respectively – & thus aligns with recommended guidelines for the responsible use of AI.

[1] White et al. "Stabilized neural differential equations for learning dynamics with explicit constraints." NeurIPS (2023)

[2] Gilmer et al. "Neural message passing for quantum chemistry" ICML (2017)

[3] Schütt et al. "Equivariant message passing for the prediction of tensorial properties and molecular spectra." ICML (2021)

[4] Kidger et al. "Deep signature transforms" NeurIPS (2019)

[5] Krishnapriyan et al. "Characterizing possible failure modes in physics-informed neural networks." NeurIPS (2021)

[6] Guan et al. "Widening the pipeline in human-guided reinforcement learning with explanation and context-aware data augmentation." NeurIPS (2021)

[7] Ghorbani et al. "Flee 3: Flexible agent-based simulation for forced migration." Journal of Computational Science 81 (2024)

[8] Geaves et al. "Integrating irrational behavior into flood risk models to test the outcomes of policy interventions." Risk Analysis (2024)

[9] Bardoscia et al. "The impact of prudential regulations on the UK housing market and economy: insights from an agent-based model" Bank of England Working Paper (2024)
Thank you for your response.

We are a bit unsure about what your comment about domain expertise is referring to without any additional context. We have nonetheless taken an educated guess as to your concern and address this below (TL;DR now; detailed response to follow).

### TL;DR

1) **Our reliance on domain knowledge is minimal** relative to some areas of interest to the NeurIPS community. Our assumption of a fixed $\tau$ map is entirely reasonable given that **modellers always know why they are building a model**. (Why would they build a model if they _didn't_ know this?) We will in any case include several detailed examples for how to choose $\tau$ in different settings; we include some of these examples below. (We believe this would also help to address the limitation listed by Reviewer `fR4E`.)

2) The principles of **keeping humans in the loop**, of **incorporating human knowledge into ML pipelines**, and **adequately evaluating a model's objectives prior to deployment**, often appear in guidelines on the **responsible use of AI** in decision-making (e.g., Principles 1, 3 & 6 of `The Institute for Ethical AI & Machine Learning`'s "Responsible Machine Learning Principles"; "Assessment and Accountability" in `The Center for AI and Digital Policy`'s "Universal Guidelines for AI"). Our assumption that there exists a domain expert in the loop who informs $\tau$ is therefore **consistent with the responsible use and development of interventionally consistent surrogates** and in this sense is a strength of our approach.
Dear authors, thank you for the rebuttal and the new experiments you mentioned in the global rebuttal. 

I still think that, although the topic is relevant for the ML community, there are strong limitations in its applicability by practitioners without strong domain expertise.
(2/2)

## Unsuitability of [R3.1]

> ...you would expect an experimental comparison which is also missing.

Suppose we attempt an experimental comparison by applying [R3.1]'s method (herein TCR) to our epidemic case study. The reviewer agrees that we should do so by "instantiating as many reduced models as the variables of interest". In our epidemic case study, there are $3 T$ such variables – each of the $S_t, I_t,$ and $R_t$. 

**TCR's first limitation**: the population size in our model is fixed to $N$, such that $S_t + I_t + R_t = N$. This cannot be enforced in TCR, since TCR models $S_t, I_t,$ and $R_t$ separately (Sec. 4 of [R3.1]: "A target scalar variable $Y$...").

**TCR's second limitation**: $S_t, I_t,$ and $R_t$ are all integer-valued and bounded in $[0, N]$. This is not handled by TCR, since TCR assumes target variables are Gaussian (p.6 of [R3.1]: "...we make a Gaussian assumption on the densities...").

Suppose we proceed with TCR despite these limitations. Following [R3.1]'s experiments, a TCR model at time $t$ would consist of $\tau$ and $\omega$ maps from the $Nt$ states of the agents over time to a target variable at time $t+1$. ([R3.1]'s double well experiment targets the final position of a mass in a double well by learning a TCR from the positions & velocities of the mass at all preceding times. See e.g. Line 115, `base.py` in their GitHub repo.) **TCR's third limitation**: For high-level models with $n$ causes, the number of free parameters across all TCR models would be of order $\sum_{t=1}^T NtnD \sim NT^2Dn$, where $D$ is the number of target variables per time step. For the epidemic case study we have presented ($D=3$, $N=2500, T=50$),  **[R3.1]'s approach would require ~$\mathbf{10^7}$ free parameters** compared to the $\sim 10^3-10^4$ free parameters used in our surrogates, even when only one high level cause is adopted ($n=1$). For larger-scale models, [R3.1]'s approach would be entirely inapplicable: to find a targeted causal reduction for Covid spread amongst a few million agents over a few months, **[R3.1]'s approach would require billions of free parameters**.

## Fixing $\tau$

> The epidemiological case study... has a nonlinear (hand-crafted, rather than learned) variable map... However... the method does not specify how to find them. 

> ...crucial parts are assumed to be given through domain expertise... most of the heavy lifting has to be done by the person that runs the simulation.

We disagree that setting $\tau$ is where most *heavy lifting* occurs. Just because a set of variables have been identified via $\tau$, it does not mean the domain expert is aware of how these variables affect and interact with each other. For example, one may be aware of temperature and pressure in a physical system, but unaware of how they relate to each other. That is, **much of the heavy lifting lies in identifying the underlying mechanism** between emergent system properties. As already stated, [R3.1] avoids much of this issue by assuming linearity and Gaussianity, whilst we do not. **[R3.1] themselves** highlight these assumptions as problematic when modelling large scale complex systems:

>  ...we made Gaussian approximations and used linear 𝝉 and 𝝎 maps. While this has clear benefits, this may be too limiting for some complex simulations...

Also, the experiments of [R3.1] rely on expert knowledge, just in a different way. In each experiment the appropriate number of causal variables are chosen to model the phenomena of interest:

- Double well: 1 variable capturing the mass' momentum at the well's apex.

- Mass Spring: 2 variables representing independent shifts in the $x$ and $y$ directions.

Choosing the right number of causal variables is challenging for large-scale complex systems, and it is unclear how one would do so without knowledge of macroscopic properties. Moreover, [R3.1] employs a regulariser to ensure that all causal variables are used by $\tau$ and $\omega$. Thus, if too many variables are specified, TCR may learn a less effective and interpretable map. We have no such issue. Lastly, note that using a $\tau$ informed by experts ensures the surrogate is **easy to interpret** in terms of macroscopic system properties. This isn't guaranteed when $\tau$ is learned.

(To be clear: we assume $\tau$ is given, but we learn $\omega$.)

## Generality of Approach

One only requires a $\tau$ map and a surrogate family to run our approach. Moreover, the surrogate families used in our SIR experiments are very generic. This is evidenced by our ecological example, which relies on the same families. In general, it's easy to specify a surrogate family that meets our assumptions. Many complex systems are studied via ODEs/system dynamics models that can be used as a basis. (This is how some of our classes of surrogates are motivated.)
(1/2) 

Thanks for your response. We provide a summary TL;DR before providing a detailed response in our next comment.

## TL;DR

1) **[R3.1] is entirely unsuitable** for our setting:

    a) **[R3.1] does not handle multiple interdependent target variables**;

    b) **[R3.1] does not handle discrete or bounded target variables**; and

    c) **[R3.1] scales extremely poorly** requiring $\sim 10^7$ trainable parameters for our epidemic case study, and _billions_ of parameters for more realistic simulators.
In contrast, **our approach does not suffer these limitations**.

2) We disagree that fixing $\tau$ does the "heavy lifting": **knowing the concepts you care about doesn't mean you know the mechanism that relates them**. Moreover, we believe that fixing an interpretable $\tau$ is easier than setting opaque hyperparameters to learn a potentially uninterpretable $\tau$, as in [R3.1].

3) **Methods for nonlinear and non-Gaussian abstractions are recognised by [R3.1] _themselves_ as important**. Our work has developed such a method contemporaneously with [R3.1].

Whilst we believe our work is contemporaneous with [R3.1], we are still consulting the ACs about the evidence we can provide without breaching review policy. Thus, we will address the point regarding arXiv in a later comment.
Thank you for providing a detailed rebuttal to my review. I went over the rebuttal, the other reviews and had another look at the paper and will try to comment on the rebuttal below.

## On the contributions with respect to prior work

From the section on contemporaneous work from the [Call for Papers](https://neurips.cc/Conferences/2024/CallForPapers):
> For the purpose of the reviewing process, papers that appeared online within two months of a submission will generally be considered "contemporaneous" in the sense that the submission will not be rejected on the basis of the comparison to contemporaneous work. Authors are still expected to cite and discuss contemporaneous work and perform empirical comparisons to the degree feasible.

Since [R3.1] appeared online in November 2023 (see the [arxiv version](https://arxiv.org/abs/2311.18639)) it is not considered contemporaneous work for this submission, as the submission deadline was in May 2024.

The paper's contributions, as summarised in the first part of the paragraph starting in L 49:
> To address this, we build on recent developments in causal abstraction [...]. We view the complex simulator and its surrogate as structural causal models [...], and propose a framework for constructing and learning surrogate models for expensive simulators of complex socio-technical systems that are interventionally consistent, in the sense that they (approximately) preserve the behaviour of the simulator under equivalent policy interventions. This perspective enables treating the surrogate model as a causal abstraction of the simulator.

The points in the part above have been covered in [R3.1] (with some minor differences that were mentionned in the rebuttal and which we will discuss below). Such that the statement in L58:
>Our approach establishes, for the first time, a connection between complex simulation models and causal abstraction, and a practical approach to learning interventionally consistent surrogates for complex simulators.

is incorrect (same goes for L369).

Now we can consider the merits of the paper regarding the second part of the contributions (L55):
> We motivate our proposed methodology theoretically, and demonstrate with simulation studies that our method permits us to learn an abstracted surrogate model for an epidemiological agent-based model that behaves consistently in multiple interventional regimes.

and as an extension of prior work.

The epidemiological case study in Sec. 5, does encode more complex causal structure and has a nonlinear (hand-crafted, rather than learned) variable map from the simulation to the surrogate.
However, as discussed in the limitations section of my review, the method does not specify how to find them.
Prior work [R3.1] defines a method to learn the maps between variables and interventions under some simplifications (as outlined in the rebuttal).
This submission, however, generalises this to exactly one case study (or two if you count the results promised in the rebuttal) and does not provide a method that is directly applicable to the wider ML/simulation community.
The crucial parts are assumed to be given through domain expertise and/or not learned by the method.
The part that is general about the proposed approach (Sec. 4) vague and most of the heavy lifting has to be done by the person that runs the simulation.

Furthermore, in order to consider this an extension of prior work you would expect an experimental comparison which is also missing. The comparison could be done by "instantiating as many reduced models as the variables of interest" as mentionned in the rebuttal or choosing one variable to compare against.

I commented above on the theoretical results in the initial review.

Therefore, besides some of the main contributions having been covered by prior work, I think the limitations outweigh the contributions of the proposed approach when viewed as an extension of it.

## Questions

Q1: Couldn't you always consider the parameters defining the causal mechanism as a separate variable and intervene on this separate parameter variable? Suppose your soft intervention changes the mechanism from the original one to a member of a family of mechanisms and the parameter variable acts as a "mechanism selector". In that case, wouldn't the approach be applicable?

## Conclusion

I appreciate the effort put into the rebuttal and the clarifications provided. However, the primary concerns regarding the novelty and applicability of the proposed method, especially in light of prior work, remain.
**Refs**

[R1.1] Geffner et al. "Deep end-to-end causal inference" NeurIPS Workshop on Causal Machine Learning for Real-World Impact (2022)

[R1.2] Melistas et al. "Benchmarking counterfactual image generation" arXiv (2022)

[R1.3] Mooij et al. "Distinguishing cause from effect using observational data: Methods and benchmarks" JMLR (2016)

[R1.4] White "Maximum likelihood estimation of misspecified models" Econometrica (1982)

[R1.5] Aglietti et al. "Causal Bayesian optimization"  AISTATS (2020)

[R1.6] Lee and Bareinboim "Structural causal bandits: where to intervene?" NeurIPS (2018)

[R1.7] Lattimore et al. "Causal bandits: Learning good interventions via causal inference" NeurIPS (2016)

[R1.8] Bilodeau et al. "Adaptively exploiting d-separators with causal bandits" NeurIPS (2022)

[R1.9] Angione et al. "Using machine learning as a surrogate model for agent-based simulations" Plos one (2022)

[R1.10] Arumugam and Van Roy "Deciding what to learn: A rate-distortion approach" ICLR (2021)

[R1.11] Arumugam and Van Roy "Deciding what to model: Value-equivalent sampling for reinforcement learning" NeurIPS (2022)