Interventionally Consistent Surrogates for Complex Simulation Models
Authors: Joel Dyer
Nicholas George Bishop
Yorgos Felekis
Fabio Massimo Zennaro
Ani Calinescu
Theodoros Damoulas
Michael J. Wooldridge
Venue: NeurIPS 2024 poster (2024)
Keywords: agent-based model
causal abstraction
complex simulator
surrogate model
Primary area: machine_learning_for_social_sciences
TL;DR: We propose an approach to learning interventionally consistent surrogate models for complex simulators using causal abstraction.

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.

Interventionally Consistent Surrogates for Complex Simulation Models
Joel Dyer* University of Oxford
Nicholas Bishop University of Oxford
Yorgos Felekis
Fabio Massimo Zennaro
University of Warwick
University of Bergen
Anisoara Calinescu University of Oxford
Theodoros Damoulas University of Warwick
Michael Wooldridge University of Oxford
Abstract
Large-scale simulation models of complex socio-technical systems provide makerss with high-fidelity testbeds in which policy interventions be evaluated and what-if scenarios explored. Unfortunately, the high computational of such models inhibits their widespread use in policy-making settings. Surro- gate 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 framework for learning interventionally consistent surrogate models for large-scabl complex simulation models. We provide theoretical results showing that our proposed ap- proach induces surrogates to behave consistently with high probability with respect to the simulator across interventions of interest, facilitating rapid experimenta- tion with policy interventions complex systems. We further demonstrate with empirical studies that conventionally trained surrogates misjudge the effect interventions and misguide decision-maker towards suboptimal interventions, while surrogates trained for interventional consistency with our method closely mimic the behaviour of the original simulator under interventions interest.
Introduction
Large-scale, complex simulators powerful tools for modelling distributed socio-technical systems and emergent phenomena across application domains, including the social sciences [Wiese et al.) 2024), epidemiology (Kerr al. 2021]. and finance [Cont] 2007]. Many such systems consist of multitude of autonomous, interacting, and decision-making agents, whose individual behaviours and interactions can be captured more readily and at higher degree of fidelity in 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 testbed for experimentation with policy interventions. In economies, 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 nonn-phaarmacuuticll 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
joel dwwrrccees...o ox, ac uk
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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for single forward simulations (Jagiella et al., 2017 Fadai et al., 2019 Wright and Davidson 2020 Heppenstall al. 202T]. Since extensive simulation studies are often required to aid secision-making with these models, such costs present a barrier to their use 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 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 downstream tasks involving experimentation with possi- ble 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, [019 Zennaro et al. [023a]. We view the complex simulator and its surrogate as structural causal models [Pearl] 2009], and propose 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 practical approach learning interventionally consistent surrogates for complex simulators. Our work provides an avenue for researchers modelling complex socio-technical systems 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 experi- mentation 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 (World 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.
2
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.
2.1 Structural causal models
SCM a rigorous model describing a causal system: Definition (SCMs (Pearl 2009).. structural causal model M a tuple (X, U.F, where:
X [x is a finite set of endogenous random variables X; each with domain dom
2

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U = (- is finite set of exogenous random variables, each with domain dom[U,] and each associated with an endogenous variable;
== is a finite set of measurable structural functions, one for each endogenous variable
defined as ft dom(PA((..) x dom[U,] + dom[Xi], where PA((;) C [X] X,,
(U) is joint probability distribution over the exogenous variables factorizing as Th (U.))
The model M is associated with Directed Acyclic Graph (DAG) 9M = (V,8) where the set V of vertices is given by X UUU and the set & of edges is given by {(S}, X;) Sj E PA(X;) U
Definition [ conforms to the standard definition of 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 (U) over the exogenous variables can be pushed forward over the endogenous variables, defining the probability distribution (M) # (IP (U)) Joint distributions PPM (S) can then be defined for any subset 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 hard interventions, in which fixed values are assigned to subsets of endogenous variables:
Definition (Interventions PPearl 2009)). Given an SCM M, S and set of values realizing S, an intervention do(S S), is an operator that replaces each function Sc associated with S; with constant St.
The intervention + do(S s) induces new post-intervention SCK, M, = identical to the original one, except that in F, the functions f. are replaced with the constants 8j. The probability distribution of M, is computed as IP, M. (X (X\S). Graphically, the intervention transforms the DAG of M by removing incoming edges in each variable S:-
We use denote a of feasible interventions on the SCM M that are relevant to a policymaker. Intervention sets are equipped with a natural partial ordering: let = (S s) and i2 (T = t); then 1 3 12 iff (i) S S T, and (ii) for each S, Ti holds s; t,; informally, 41 intervenes on subset of the variables that (2 intervenes on, and it sets the same values as
2.2 Complex simulators as structural causal models
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 endogenous variables, deterministic dynamics into structural equations, and factoring sources of randomness into exogenous variables. The following example illustrates how common ABM from epidemiology can be cast as a SCM.
Example (Spatial SIRS ABM). We consider a epidemic model state of each agent can be 0, 1, or respectively, indicating that the agent disease-free and susceptible to infection, infected, agents at discrete time step i [0, I'] is written as x; E {0, 1, 1,2] where is the total number of m) for integers 1. S The states Xt,n of
inn-ebb---ineeceed--eeveeeeeer I, x T, lattice of cells, each of which represents one of N T,² agents. The
(SIRS)
is recovered from recent infection. The infection status of all
simulated time steps, and [l, m] = {1, + l,..., each of the agents n E [[1, N] are updated synchronously as follows for E [0,T-11 1]:
(UI) X = 0, then X = with probability
(t,n(ae+1)
(1 a441 E [x
where Nn, is the von Neumann neighbourhood for cell n; else remain susceptible.
(U2) Xf then X 2 with probability B441i else remain infected.
(U3) If Xt,.. 2, then Xt+ = O with probability 16s41 else remain recovered.
In the above, O. (as, (t) [0, 113 are the model parameters determining the transition probubilities between states. While these may vary over time, the simplest case consists of assigning all 0, the same vector,
0: =v
Vt [1,
(2)
3

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The model is initialised by infecting each agent in the model at initial time t = O with probability Io [0, 1]. The value of to for any forward simulation of the model can be chosen by drawing random variable from some distribution on
[0. 1| and setting
10 a.
With this model in place, lockdowns over some time period t t1 + A of length A 2 can be modelled (crudely) by setting 0 01 (0, () B, [0,1]. To express this ABM SCK, we define the following:
Endogenous variables These consist of the variables of interest that may be set by the policymaker: Iu. (xclesisr and {0thssgT.
Exogenous variables The model as described above is initialised randomly according to a, and a collection Uo = (1Ssn) of N random variables distributed as (4 1). the nah of which helps determine whether agent n. is infected at time t = O. Similarly, further collections u,, E [1, T] of 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 us for I. [O,T].
Structural equations Equations 2 and 3 respectively, define the structural equations fo, and St for the endogenous variables 02 and To- The structural equation fxa,, for each Xn,n: [1, N] can furthermore be written as
Xo.n = Sxo,n (Uo Io) [[ < <o0--
Finally, update rules (UI)-(U) can be written in the following way for t E [O.T- - 11]:
Xt+1,n = [xit1,. (0++1: 11+1, xx,))
=I = = 0] [ut4i, < Pt.n (a 1 (x,n =
Bitil)
+ 21 [x,, = 2] (1- (1- < <+1l))
(5)
Distribution over exogenous variables The (random) behaviour of the exogenous variables is fully specified by the distribution over and v, along with U(0, 1) distributions over the Un
The underlying graph The DAG corresponding to this SCM is shown in Figure 7 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 0, as do (0, 3)) for some 3,y E [0, 1], while in the observational regime the O₂ will all be assigned the same value.
We emphasise that the above example in- tended only to illustrate that complex simulators, such as ABMs, can seen as SCKs; explicitly rep- resenting a given simulator an SCM in the example above not required in the sequel.
U3
a
1o
X
X2
X3
2.3 Causal abstractions
O
O₂
0
Beside 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 sys- tem at different levels of granularity. The notion of exact transformation formalizes this relation, providing framework to relate complex models, such as ABMs, simpler top-down models, while preserving causal structure.
Figure 1: The directed acyclic graph induced by the structural causal model for the spatial SIRS agent- based model for T time steps.
Definition 3 (r-W Exact Transformation (Rubenstein et al. 2017). Given two SCMs, M and M', with respective intervention sets I and I', a T-W transformation is pair ()) consisting of

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+ dom[X'] and surjective, order-preserving map w : I - I' An exact T-W transformation a T-W transformation such that
map T dom[X] X
(I (P..) = PM Vi EZ.
(6)
An exact 7-w 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 intervening via i and then abstracting produces the same result as abstracting first and then intervening via w(t). The map r describes corresponding states in each of the models, while the map w describes corresponding interventions in each model. Whenever the map is clear from context, we herein shorthand the pushforward measure T# (P M,) as P##
M.
An exact 7-W transformation between the SCM M underlying a complex simulation model and the SCM M' underlying the candidate surrogate model would (a) certify that the surrogate pre- serves the causal behaviour of interest, guaran- teeing interventional consistency when policy- makers study interventions through the surro- gate, and (b) allow to interpret the emergent causal structure of the simulator through M..
M
PM.
w())
M'
P.M
Figure 2: Computing T# (P.M) corresponds to moving right, then down, in the diagram. That is, running the intervention in a base model M such as an ABM. Computing IP MM corresponds to moving down, then right. That is, running the intervention w()) in an abstracted model M such as surrogate. If (r, w) is an exact transformation. then the diagram commutative for all interven- tions. That is, T# (TP MP) P MLL for all i E I.
3 Abstraction error
is often unrealistic to assume that an exact 7-W transformation exists between a complex simu- lator and its surrogate A more pragmatic goal is to find approximate abstraction [Beckers et al. 2020 from the simulator to the surrogate. We therefore define the abstraction error:
Definition 4 (Abstraction error). Let (T,w) be a T-W transformation between two SCMs M and M' with respective intervention
and I'. Given statistical divergence between distributions, and a distribution over the intervention set I, we define the abstraction eπror as follows:
(M, E.- [d (TM (FMM): PMC )
A r-w transformation is a-approximate for some a E R20 if do (M, M') S a
A T-W abstraction with low abstraction error im- plies that Tw (PP, MM. is close PM in expec- tation with respect to the interventional distribu- tion 77. If the Tw (PPP. TP P. is zero for all interventions E I. then (T,w) is an exact transformation (see Figure
MM
P.M. Lr
T# (PM.)
dr
wi() M PM
Definition 4 differs from previously defined tions of abstraction error in the causal abstrac- tion literature. Whilst BBeckrs/ al. 2020) em- ploy a maximum over interventions, we instead take an expectation over a fixed interventional distribution 7). 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 specifi- cation of 7) one may implicitly favour surrogates which perform well with respect to interventions of high interest. Further discussion is provided in Appendix B
Figure 3: The abstraction error compares the distri- butions T (PMM,) and P. FM... for each intervention i using the divergence dr, indicated by the red dotted arrow. If the divergence is zero then we recover the commutative diagram in Figure 2
4 Method
The definitions of abstraction and abstraction error provide us with framework for learning sur- rogates, and, in the remainder, we assume that the base model M is implicitly represented by
5

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simulation model of a complex socio-technical system. Our goal is then to identify surrogate model which is interventionally consistent with this simulator. Specifically, from a set of candidate surrogate models M., we seek surrogate and a 7-w transformation that minimises the abstraction error.
To proceed, we assume that 21 is parameterised family 21 {M t vD of differentiable surrogate simulators with tractable probability mass or density function qy. Here, MM denotes the causal model induced by a surrogate whose structural equations are parameterised by e), and y denotes the set of feasible parameter values. Such 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 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 T# (TPM), amounting running the simulator and applying T 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.I We thus assume that the map т, which defines the aggregate, emergent quantities of interest to the policymaker, is pre-specified.
Hence, to find an appropriate T-w transformation, we need only to identify an intervention map w between I and I'. For computational tractability, we select w' from a parameterised family Ω := {wo Φ } with parameters e ranging over the set Φ. For example, ó may be the weights of a neural network. We then select o* and ai* jointly by minimising di (MM, over Qa x 22 Since each element of DR has a differentiable and tractable distribution, convenient choice of discrepancy d is the Kullback-Leible (KI.) divergence, such that out problem becomes:
diP
0²,0" arrg min dr (M,M with d = E, dE
log dP M wP(()
M.
(8)
The KL divergence can be minimised using Monte Carlo estimates of the gradient
G(d, Voi d (M,
1 B
Σ
Now -oo log (uoe (y((0)
where ((6) 1) (b) Tw (PM,,u)) Lon is the probability mass/density function for M (()) and B 2 is the size of a batch drawn from R 2 B training examples from the joint distribution over the (0) and y()), Once (Ø',.²) (6*) has been selected, we may generate data from the macromodel for Algorithm summarises the training procedure.
ABM intervention by sampling from P M
A Algorithm 1: Summary of the training procedure.
4.1 Theory
Input: Budget R: batch size B E [1, R - 1]; ABM M: intervention distribution n); surrogate family D abstraction map family 2
Definition is closely related to exact trans- formations:
Proposition 1. Let be an interventional
Result: Trained surrogate and abstraction map parameters, 4+* and o²
distribution, d be a statistical divergence, and (T, w) be T-W transformation between SCMs M and M'. If T-W is 0-upproximate (dr, = 0), then we have a-almost- surely
Set D = 0:
for r 1 to R do
17, x()) ~ P.M.,n) D (20)) ((x(1)))
Sample (())
T# (PM,) PMiLas"
end while not converged do
The proof is in Appendix D.I In particu- lar, when I is finite and 17(1) then any 0-approximate T-W transformation is an exact 7-w transformation between M and M'. This motivates our own choice of loss
Sample minibatch {((()) T (xx()))) uniformly from D;
0 Vi E I,
Take gradient step in o v using Equation 9
end

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function: minimising Equation 8 induces & and s/ to produce a surrogate that behaves the same way as the simulator under interventions of interest.
Definition employs an expectation over an interventional distribution 7/- As a result, even when the abstraction error is low, there may still be large discrepancy between T (IP M,) and P. for some fixed intervention E I. Proposition 2 provides an upper bound on the error associated with any intervention sampled from lt when d is the KL divergence and the simulator state space is finite:
Proposition 2. Let d be the KL divergence and CE, Ey (PM.) | log (w(s) (Y) denote the cross-entropy of P. Misi with respect to T# (IP MMP. Assume dom[X] is finite. Then for all e > O.
Pn (d TH# (PM,) PM 26 E [CE]
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, s and 6, are found by minimising Equation g
5
Case study
U3
Here, we outline a case study in which we learn interventionally consistent surrogates for the spatial SIRS ABM from Example[ allowing us experiment more rapidly with policy in- terventions 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 To E [0, 1], 0, E R2 for 1 EE [1, and ýr E {(a,b, | a. b, E No, b + N) for [o, T]. where a. b. denote, respectively, the number of sus- ceptible, infected, and recovered individuals in the population. The DAGs underlying the SCMs each of these families are as in Figure 4 and the three families differ only in the form of the structural equations mapping from to and Oo.: to the ýr. Throughout, we let 9ºd be Multinomial emission distribution and & be trainable parameters of these structural equations.
1o
yo
Y1
Y2
Y3
0
Figure The DAG induced by the SCMs correspond- ing to the surrogate families for T = 3.
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 g². Here, sb =
Surrogate family consists of latent ODE-RNN (LODE-RNN), where we run a recurrent network (RNN) with parameters vp over the output of the SIRS ODE. The RNN outputs the class logits of 9dh Surrogate family consists latent RNN (LRNN) constructed by running an RNN with trainable parameters v over the 0,, and the output of the RNN at each t E [o, T] indexes the class logits of 9d)
Given Ô 0u., Jo. these surrogates enjoy tractable likelihood functions, which factorise as (YooT 01.7, Io) = (5o Io) II 8th (ý, θ 0 Io).
Interventions & the r-W transformation. Denoting
Iv do (01.T v, a),
(10)
by do (0 = 0G, = 0.001 = = v (0, = a),
we define two subsets I Timit U Isnit, lock of interventions for the ABM:
Zinit = (wv (v, E [0, 1]2} and Iniit lock (Ivaad (v.a,t)) tr) E [0, [5, 10]]. (11)
*Code for reproducing the experimental results is available at bcttp:///ithhr https://eithub..com/oooleie com//oelnndyrr/ neurips_ics4cs:
7

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Table 1: Metrics for interventionally (I) & observationally (O) trained surrogates on interventional
(I') & observational (O') test sets (median first quartile
third quarille from 5 repeats). Bold denotes best performance.
Test
Model Train
LRNN
LODE-RNI
LODE
0
AMSE (x10-1)
3.483.21
49.4
3.35
18.5
8.155.06
22.4
I'
ANLL (x 10%))
2.09
20.8
1.99
8.40
4.01
10.0g.91
9.91
AMSE (x10-l)
4.134:11
2.95
3.593.51
2.52
18.4
4.364
40
O'
32
ANLL (x10²)
2.222:73
1.64
1.86
1.43
53
7.637.52
2.15
The first of these is 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 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 ty E [5, 10] with duration equal to 5 time steps, and (c) the value of ty. Similarly defining
L,.a do (0L: V, To =a).
(12)
33, do (0zdi-1 0i.-6.r v, Oiniis (0, v, =
we define I' = Init U UTinit lock for the surrogates, where, letting ID = =R x [0. 1]. we have
Lnit = (66.6 (V, 0) D) and Tock, lock = (6a,a, | (V, a, ()) E 5,, 10]}.
(13)
The map is taken to map: 0; identically to 0g for each E [[1, 7]: the microstate x, of the ABM at each time step to the ý, through an aggregation map that counts the number of agents in x, in each of the three states (susceptible, infectious, and recovered); and the initial proportion lo of infected agents in the ABM identically to Jo. Further, for a neural network f [0, +
R3
20. we take
w : bva + (oo())
by,0,4 + Ljocr)an
(14)
The benefits of training for interventional consistency We use Algorithm to jointly learn the parameters Φ, y of the surrogates and the map wh described above two different ways: training the surrogate models with 1) taken to be uniform distribution ((Imn)) over Iuits which entails comparing the behaviour of the surrogate and ABM without lockdowns different parameters; and training with 7) instead taken to be a uniform distribution (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 Appendix D 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 dataset I' = (( 3 of size R' = 1000, generated as (6') ~ 7 = 4 (I), Yo. ~ T# (PM,,,)) Specifically, we inspect the average mean squared
(r')
error (AMSE) between trajectories from the trained surrogates and (r'), 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 77 = (Intt..
[ 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 experiments with the surrogate. We also report minor drop in observational consistency when training with data from the combined intervention set I instead of Imit. which can be explained by the overfit of the doseevvtiinaalyytrainnn trained model on the observational distribution. We also observe
included during training if the policy-maker intends to perform policy
8

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LODE RNN (T)
LODE
0.70
0.50
0.00
Time
Figure 5: Example trajectories from the ABM (middle) and the LODE-RNN trained interventionally (left) and observationally (right). A lockdown is imposed at the dashed vertical line. Solid (resp. dot-dash) lines show trajectories under (resp. without) the lockdown. The transmission-inhiee -nniiiiiinnniiiiiiing effect of the lockdown is vastly underestimated in the observationally trained surrogate, while the interventionally trained surrogate accurately predicts a reduction disease transmission.
that the LODE-RNN which combines the "mechanistic" SIRS ODE with 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 example of possible negative consequence of failing to train 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 time 4 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-RNT 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 of 5 training repeats,
observational LRNN predicts that no lockdown was the best intervention 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 the best intervention, and only the interventional LODE model predicts that no lockdown is not the worst option (in only out of 5 training repeats). This highlights the potential importance training surrogate models for interventional consistency when their purpose is to help inform downstream decision-making tasks. Furthermore, this suggests that 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.
Related work
Surrogates are often used to expedite dimulation-basse 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 al. 2023 artificial neural networks (Aniruhh
et al. 2022 De Leeuw et al.) 2023]. support vector machines ten Broeke et al. 202T), kriging (aalle
and Yildzzgguu 2014]. and mixture density networks (MDNs) [Platt [002]. [P22)) Our experiments also rely on established machine learning methods to construct surrogates; for example, our LRNN surrogate family resembles that of Platt 2022). which MDNs are used to approximate an ABM's transition density. However, in such works, the cansal/ineerentionaa consistency of the surrogate with respect to the simulator and policy interventions of interest not considered. 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 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) extended this work by proposing stricter definitions of causal abstraction, and in Beckers et al. [2020), where approximate abstractions are introduced to account for uncertainty
and simplification. Causal abstraction found practical application in Geiger et al. [202] for learning
interpretable neural networks. Rischel and Weichwald [202T] discusses an alternative category-
theoretical definition of abstraction; this was used to learn abstractions transfer data between
models at different levels of abstraction in Zennaro et al. [2023a]. Further related work includes a
multi-marginal Optimal Transport solution to the abstraction learning problem (Felekis et al. (002). as well as constructive abstraction learning in neural causal models (Xia et al. [2023] and cluster DAGs
(Anand et al., 2023). However, none of these approaches reduce the state space of the SCM or the cost of simulation, our approach does.
7 Conclusion
We propose a rigorous framework for learning interventionally consistent surrogates for complex sim- ulation models, formalised with casual abstraction. This 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 maximum as in Beckers et al. 2020); this produces computationally tractable optimisation problem, but introduces the possibility that one or more interventions is captured poorly by the learned abstraction map, even for low abstraction error. In our experiments, we have assumed surrogate models with tractable and differentiable density functions, permitting us use 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 Taaaak and Vanden -ijnden 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/DG 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 required from the base model [Lattimore et al. 2016,] Bilodeau et al., 2022]. However, it is unclear whether not applying the do-caleulus 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 generically applicable.
SCM, making it
Acknowledgments and Disclosure of Funding
JD. NB, AC, and MW acknowledge funding from UKRI AI World Leading Researcher Fellowship awarded to Wooldridge (grant EP/W002949/I). MW and AC also acknowledge funding from Trustworthy Integrating Learning, Optimisation and Reasoning (TAILOR), a project funded by European Union Horizon2020 research and innovation program under Grant Agreement 952215. YF: This scientific paper was supported by the Onassis Foundation Scholarship ID: F ZR 063- 1/2021 2022. TD acknowledges support from UKRI Turing AI Acceleration Fellowship [EP/V02678X//].
10

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A Assumptions Underlying Markovian SCMs
Definition ! implies the standard assumptions of (i) acyclicity of the DAG GM 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 faihffllees, guaranteeing that independencies in the data are captured in the graphical model Spirtes et al. 2000)]
B Other Notions of Abstraction Error
As discussed in Section
3
Definition 4 is closely related to the notion of abstraction error introduced
by [Beckers et al. 20200 In contrast to Definition 4 Beckers et al. 2020 employ maximum over the intervention set Z instead of an expectation. Hence, the abstraction error introduced by Beckers
al. (2020) may be viewed as worst-case version of Definition 4
In addition, Beckers and Halpern 2019] assume the intervention map w can be implicitly defined by the map 7, and require the abstraction map 7 to be consistent. That is, the image of I under the intervention map induced by T must equal T', Since we do not couple the maps T and w we enforce no such condition. Additionally, [Beckrrs et al. [2020) enforce surjectivity of 7. Since this makes no practical difference in a surrogate's s use in downstream tasks, we dispense with this assumption.
Alternative notions of abstraction error have been introduced by Zennaro et al. [203bb]. building upon the notion of exact transformations introduced by RRschel 2020J. Such alternative notions of abstraction have recently been used to define transfer learning protocols between SCMs related by an approximate causal abstraction in Zennaro et al. [2024]. 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 1) instead of maximum, and we leave this as a direction for future work.
C Additional Related 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 [0033 and minimum simulation distance Grazzini and Richiardi 2015). We refer the reader to Dyer et al. [2022] 2024] for thorough surveys. Unlike our framework, surrogates trained for the purpose parameter estimation do not typically account for interventional consistency explicitly.
More generally, our framework bears similarities latent space modelling of Markov decision processes (MDPs) [Gelada et al., 2019), wherein one attempts to learn a smaller latent MDP from 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 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 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 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 focus accurate simulation from interventional distributions. Our approach
15

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is therefore more tailored to large-scale and realistic nonlinear simulators. In contrast, the method presented in Kekič al. [0233 becomes impractical for large-scale models.
C.1 Examples of T Maps in Real Modelling Scenarios
We provide here some practical examples, beyond the two studies present, that illustrate how the т 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. т would therefore be defined by counting the number of agents in each of these states at each location, i.c. T,a(ZT)
ΣacAA I[agent a has demographic features and is location time t] where | indexes locations, d are demographic features, 2 is the state of the simulation at time t, A is the set of all agents, and is the indicator function.
2. Consider the model of flood risk mitigation behaviours proposed in Geaves et al. [0244]
which models how households decide to take precautions to protect themselves from floods in high flood risk areas. The modellers are interested 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). ). 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.] (204]). in which households consume goods, provide labour and invest 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 inflation rate, unemployment rate and real interest rate that are of interest to the modellers. 7 would therefore be defined by standard macroeconomic formulas for these quantities.
D Proof
D.1 Proof of Proposition I
Proof. By non-negativity of the divergence d we have (T (TT((PM))) (PM,), PMS..) 2 0 for all E I. Hence (M. corresponds to expectation over non-negative random variable. Since this expectation is equal to zero, we conclude that d (T# (I M,), F PM) = 0 almost surely with respect to the distribution 17. Positivity of the divergence d then implies that T4 (PM,) = PM almost surely with respect to the distribution
L
D.2 Proof of Proposition] 2
Proof. Using
Markov's
inequality and the fact that drwn (M.. =
Emng d
TH (PM.), M
Py ( (d((T((MM)) (PP.
22) drwe (M.
Since we have finite domain, the likelihood functions associated with (a) the pushforward measure of the ABM under and (b) the surrogate macromodel can be written as probability mass functions, whose logarithms are non non-positive. Since we have assumed I M w)
< T# (F,.), we have that
0 < log ( (Y) oo for any Y T (P..),, and therefore
0 < Erwn [CE,] <
(15)
16

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c++1
ODE Solve
U++1
MN
V++1
2
Figure 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, Zz-1, is computed via ODESolve. Then, 2 serves the logits for multinomial distribution from which yı is sampled. This sampling procedure is denoted by MN the diagram. The exogenous variables required to reparameterise the multinomial distribution during sampling denoted by ii,
We also have that
(PM.) 50 + Euwn 1- ) S 20,
(16)
where lil, (PM.) ) is the entropy of the probability mass function associated with T (P,,), and that
(PMM))) My Mi..) -HI (Pm.) CE2
(17)
d (M.M*) Bunn (P,,) ) + Euny [CE,
S
E [CE]]
(18)
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 (IP (P,,)) for an arbitrary ABMs, Hence
evaluating the density associated with TH
( Pn (a (MTT(PM)) M (oco)
)
E En [CE,]
2
(19)
E Further Experimental Details
As described the main text, the three surrogate families we consider have SCMs whose corresponding DAGs can be drawn in Figure In this section, fully specify the corresponding SCM for each surrogate. Furthermore, for each surrogate, we provide details on the procedure used to train the parameters d and 6, which respectively describe the structural equations of each SCM and their corresponding intervention map w.
E.1
The LODE Surrogate Family
To construct a set Di of probabilistic SCMs, we define latent neural ordinary differential equation (LNODE) based on the classical SIRS ODE system. The SIRS ODE system takes the form
d$, dt
di. dt
RR - a,I. dn, dt R..
aiI. - BrIe.
(20)
where 0, (ār, 3c, 7c) E RR3 are the ODE parameters and z, = Is, E [o, 7] is the ODE state, where S is the two-simplex. Note that z, represents the proportion of susceptible, infected and recovered individuals in the population according the SIRS ODE. Whilst the parameters 0, 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 v E IR3 to all 0, when interventions applied:
θ.v Vt E [O,T].
(21)
In other words, Equation (1 describes the structural equation Jo, for 0,. Practically speaking, the choice of v is inconsequential, as we can model any change to 0 as an intervention. Given 0,, the ODE state z, evolves according to the following rule:
ODESolve(Z-11,0) 0z). t [1,7]]
(22)
17

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0,- Z1
ODE
h
hr+1
O1+1 u++1
RNN
FF
MN
Ye+1
Solve
Z
Figure 7: A schematic representation of the LODE-RNN surrogate family for single time step. First, the output of the SIRS ODE for the next time step, Z1+1, is computed via ODESolve. Then, Zt+11 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, to h,+1- The updated hidden state is passed to feedforward neural network (denoted by FF in the diagram), which computes the logits 0,+1 for a multinomial distribution from which Ve+1 is sampled.
between times and t. In our experiments, we compute this using a Euler scheme with step size At = 1. The initial state of the ODE is taken to be = (1 To. Io. 0). One may change the initial state through interventions on To which is modelled as an endogenous variable.
where ODESolve denotes numerical integration of System 20
Given 2,, draw the endogenous variables y; from multinomial distribution whose class proba- bilities are given by Zç. Whilst &; represents the percentage of susceptible, infected, and recovered individuals predicted by the SIR ODE, y, represents the actual counts observed by the experimenter. We write fy. (To. 01:,, ūt) denote the structural function associated with yt. where the dependence on To and O- for S is mediated by the trajectory followed by the 2r for S t, and ú, are the exogenous random variables required to reparameterise the multinomial sampling procedure on each time step.
Note that vi = 0 for this family of surrogates, and hence Dt is a singleton. For the function f comprising the intervention map w we take feedforward network with layer sizes 3, 32, 64, 64, 64, 32, A ReI.U activation is applied after each hidden layer, and sigmoid activation is applied to the final output layer. The sigmoid activation function ensures that the predicted intervention vector f (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 At = This feedforward network consists of 12,739 trainable parameters.
E.2 The LODE-RNN Surrogate Family
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 ReLU activation function which maps from the hidden state h, E RR32 of a GRU recurrent network that is passed over the trajectory 2p:T generated from the SIRS ODE (forward simulated 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 fy. (Jo, 0v.,, ū,) + ýr for each E [1,
For this model, t is the collection of trainable parameters comprising these GRU and feedforward networks. For fo, we a feedforward network with layer sizes 3, 32, 64, 32, 3, where 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 to and á combined is 13,798.
E.3 The LRNN Surrogate Family
This surrogate family makes use of the SIRS ODE model. Instead, the logits of the multinomial distributions for E [1, 7' are indexed by the outputs (o)...., OT), )r), E R33 of 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, E I2 of a GRÜ recurrent network which is passed over the sequence 0.. The initial hidden state is chosen to be ho = (1 Io. Io. O), where 0 is vector of 30 zeros. We also take ou (log(1 11 fo), log(To), -o0) which indexes the logits of the multinomial distribution at time t O. Once again, we may write the structural equations fy, for
18

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0 0.11 h
Ut+1
RNN
FF
MN
> Ve+1
h
06+1
Figure 8: schematic representation of the LRNN surrogate family for a single time step. First, the parameters 0z+1 are passed to a recurrent neural network (denoted by RNN in the diagram) that updates hidden state. The updated hidden state is passed a feedforward neural network (denoted by FF in the diagram), which computes the logits 0c+1 for multinomial distribution from which Y141 is sampled.
the Yr terms of To, O1. and the exogenous random variables ú; required to reparameterise the sampling procedure from the multinomial distribution.
Since exactly the same networks in this surrogate family in the LODE-RNN family, the total number of trainable parameters from & and & combined is also 13,798.
E.4 The likelihood function for each of these surrogate families
Having intervened on the Io and 0, with known values, the class probabilities for each multinomial distribution completely determined given the deterministic dynamics within the structural equations mapping to the yt.
E.5 Formalising the map
Taking dom [T] = JM [0. 1). dom [Xor]] XT+1 with X' = (,, and dom(O1.7) = P3
with P [0, 1|3 we define
JM x T+1 x XP - IM YT+1 P/
which operates componentwise as
T (Lo, Xo.T, .o:T) (To())), To (Xo.)), To (0 (0..T)
(23)
where
7i id,
(24)
Tr Xp:T
Σ
xx.=00
Ix,:
Ixm==2 =2
To id.
(25)
In the above, id is the identity map, and Tu acts by counting the total number of susceptible, infected, and recovered individuals in the ABM at each time step.
E.6 Further experimental details on the training procedure
All models were trained on CPU on 2022 MacBook Pro, operating on macOS Ventura 13.2.1.
Training one surrogate model on this machine took average approximately 20 minutes, amounting to approximately 600 minutes total to produce the results reported in Table [ 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 I which is used in all of our experiments.
As suggested by Figures] step.
and 6 the parameters 0 and ² are fed into the models at each time
For the LODE and LODE-RNN surrogate families, we forward simulate the SIRS ODE with an Fuler scheme with step size At 1.
19

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For all surrogates, the neural networks comprising the wh map and structural equations parameterised by i were trained with a learning rate of 10 for 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
observational and interventional training sets; these were each split 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.
E.7 Additional Case Study
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 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 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. family of deterministic mechanistic models based on discrete-time of population dynamics [see, e.g., (Saoo 2005), where (analogously to the LODE surrogate family discussed in the epidemic case study) the underlying deterministic dynamics of the population dynamics model index probability distribution at each time step (in this case, a Binomial distribution for each of the 4 species);
model
2. an LRNN family, exactly mirroring the LRNN family considered in the epidemic case study presented already;
3. and a third family considers hybrid approach, where (as in the LODE-RNN family considered in the epidemic case study) we pass a recurrent network over the underlying Lotka- Volterra- type population dynamics model first before taking the output of the recurrent network to index the Binomial distributions for each of the four species.
table for the results of this additional case study is shown in Table [ where we see that the results 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.
20

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Table 2: Metrics for interventionally (I) & observationally (O) trained surrogates interventional (I') & observational (O') test sets (median first quartile
third quartile from 5 repeats) for the predator-prey case study,
AMSE & ANIL measure ability model counts of each species over time. Bold denotes best performance.
Test
Model
LRNN
LODE-RNN
LODE
Train
I
AMSE (x10²)
1.65
3.30
1.79 .79
2.23 29
41 41.8444
201.7825.
I'
46.11
ANLL (x10*)
0.78
11.20 10.09
11.44
0.93
5.63
6 6.54 6.44
47 47.14
140.33
37.33
AMSE (x10²)
1.85
1.611.64
2.18
1.561.67
63.89
36. 36.75 25
O'
ANLL (x10*)
0.76
0.69
1.25
0.660
32. $2.11 10.36
33.90
5.49%.
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Reviews:
[{"review_id": "24bjAd77jo", "reviewer": "Reviewer_fR4E", "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.", "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?\n\nQ2: 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?\n\nQ3: 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)?\n\nQ4: 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": "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.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1: The paper addresses an important problem in the simulation of complex socio-technical, highly dynamical, systems; i.e. reducing the computational cost of running large-scale simulations when critical parameters of the model change. The proposed framework not only has potential to significantly accelerate experimentation in policy-adjacent fields, but also in other technical environments where that's the case (physics, many engineering sciences, etc).\n\nS2: The theoretical foundation is solid, building on established concepts from causal inference and abstraction. The authors provide formal definitions and proofs for their key results on abstraction error and interventional consistency, and I found the reading relatively clear (even though causal models are not my field of work).\n\nS3: The empirical evaluation on the SIRS epidemiological model provides a concrete demonstration of the method's effectiveness on what I think is a fairly representative model of the field (modulo again my superficial familiarity with such fields). The comparison between interventionally and observationally trained surrogates highlights the importance of considering interventional consistency, which neatly justifies both the paper narrative as well as the methodology.\n\nS4: Overall the connection between complex simulation models and causal abstraction feels underexplored, and this work opens up new avenues for applying such techniques across the board.", "weaknesses": "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.\n\nW2: 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.\n\nW3: 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.\n\nW4: 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1 & W4: Generally, we expect $\\eta$ to be informed by domain experts/policymakers based on downstream tasks, perhaps accounting for economic/political constraints. E.g. in a pandemic, economic constraints may preclude lockdowns of length >2 weeks, and political pressure may demand action soon; here, $\\eta$ could be uniform over 2-week lockdowns starting within a month.\n\nWe now turn to the reviewer's question on benchmarks. Whilst numerous datasets & benchmarks have been proposed in the causality literature [R1.1-3], they are typically confined to SCMs much smaller than those of large-scale simulators. Likewise, to our best knowledge, there is no consensus in the social simulation community regarding benchmark simulators, besides simple examples (e.g. the Schelling model). Further, benchmarking surrogates requires access to a set of realistic downstream tasks. For instance, to properly assess a pandemic surrogate one must know the problems epidemiologists are interested in. Summarising, benchmarking surrogate models of complex systems requires:\n- A set of large-scale standardised simulators spanning application domains.\n- A set of downstream tasks for each simulator on which surrogate models can be deployed.\n\nWhile we focus on presenting a new methodology for surrogate modelling grounded in causal abstraction theory, as we will discuss in revision, one way to devise a benchmark on downstream tasks is to check that interventional properties are preserved. For example, one can check that the internal ordering of interventions wrt their efficacy is preserved (see the Global Rebuttal).\n\nQ2: As recognised by the reviewer, the surrogate family may not fully capture the behaviour of the simulator. Note that the task of learning a surrogate may be formulated as minimising $\\text{KL}(P \\Vert Q(\\psi, \\phi))$ over $\\Psi \\times \\Phi$. Here, $P$ is a joint distribution over interventions and abstract states given by first sampling an intervention $\\iota \\sim \\eta$ and then sampling from the base model under $\\iota$, while $Q(\\psi, \\phi)$ is a joint distribution defined by first sampling $\\iota \\sim \\eta$ and then sampling from the corresponding interventional distribution in the surrogate. Given the form of this problem, classical results for maximum likelihood estimation can be adapted to provide misspecification guarantees; by leveraging [R1.4], one can show that our surrogate estimation is asymptotically normal w/ sandwich covariance & mean corresponding to the surrogate model w/ minimum KL-divergence to the perfect abstract model under $\\tau$.\n\nQ3: Our method assumes no knowledge of the simulator's SCM. This ensures our method is generally applicable as it's difficult to directly characterise the SCM of large-scale simulators. It's possible that access to the base SCM/DAG may expedite abstraction by allowing us to focus on minimal intervention sets [R1.5-6], or leverage the identifiability of interventional distributions to reduce the number of simulations required from the base model [R1.7-8]. However, it's unclear that applying the do-calculus on large causal graphs is more efficient than simulating interventions directly. We will discuss these ideas in more detail in the appendices of the paper.\n\nQ4: Many of the surrogates we consider are neural networks, thus the time complexity of sampling an interventional outcome scales linearly w/ the dimension of interventions. Moreover, our neural surrogates benefit from parallelisation within each time step, massively reducing their run-time. In contrast, social simulators often rely on pairwise interactions between potentially millions of agents that are difficult to parallelise. Thus, once trained, neural surrogates can be orders of magnitude faster to evaluate than the simulator. Precisely characterising this speed up is non-trivial & depends on the internal structure of the simulator and the surrogate. For a comparison of runtime costs in our experiments, see part (b) of our response to Reviewer fUGP.\n\nOf course, there is a training cost associated with the surrogate, but this will be amortised over downstream tasks. That is, when the collective sample complexity of downstream tasks exceeds the sample complexity of causal abstraction, learning a surrogate is beneficial.\n\nFurthermore, running complex simulators may require technical expertise & high performance hardware not widely available. Providing interpretable surrogates that run on commodity hardware reduces the barrier to entry in such cases. Note that this mirrors the release of low memory LLMs that run on commodity hardware commonplace today.\n\nW1: Please see the Global Rebuttal.\n\nW2: Our baselines are already complex wrt SOTA in social simulation (see e.g. [R1.9]). Please let us know if there are specific baselines you would like us to compare against.\n\nW3: We agree that more complex simulators are used in practice, but we prioritised clear presentation by focusing on experiments that are easy to understand, familiar to the social simulation community, & complex enough to demonstrate our method's viability. As in our response to Q1: applying our method to more complex models is complicated by a lack of standard benchmarks in the social simulation community.\n\nL1: We agree that selecting the appropriate granularity via $\\tau$ is hard, but in many cases domain experts know what emergent properties interest them. For instance, epidemiologists/labour economists/central bankers often care about the number infected individuals/unemployment numbers/inflation rates over time. In other words, domain expertise can be leveraged to select a suitable $\\tau$. We do not know of existing work that automatically selects the right granularity for abstraction; the closest we know of applies rate distortion theory to learn low dimensional representations of MAB and RL problems [R1.10-11] but do not have a causal flavour & focus on single downstream tasks.\n\n**Refs**\n\nIn Official Comment. (Too few chars sorry!)", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yP8ZwYF3B9", "reviewer": "Reviewer_fUGP", "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.", "questions": "See Weaknesses", "limitations": "Limitations of the work are reported in section 7", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "a. The paper is well-written and easy to follow. Concepts are clearly explained and equations well-commented\n\nb. Empirical results are sounding", "weaknesses": "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.\n\nb.  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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review and suggestions for improving our paper. We provide itemised responses to your two questions below.\n\n**a.** To expand our empirical assessment of the framework we propose, and to provide practitioners with a further example of how they may use our framework, we have prepared a further case study that we will include in the revised paper. We provide details on this experiment in the Global Rebuttal above. We will also include a discussion of how causal surrogates can be benchmarked going forward, such as by assessing the degree to which the surrogates preserve the optimality or ordering of interventions in the original simulator with respect to some downstream optimisation task. Please see the Global Rebuttal for further details. We believe this would also be a useful guide to practitioners, and hope that this also addresses the reviewer's question here.\n\n**b.** We will include in the revision a comparison of the runtimes of the simulators and surrogates. Briefly: in the epidemic case study presented, the `LRNN`/`LODERNN`/`LODE` surrogates each run approx. 3 times faster than the ABM; in the new experiment described in the previous bullet point and in the Global Rebuttal, the surrogates run 10-30 times faster. Obviously, the exact speed up will be different for different simulators. That said, there are additional considerations that motivate the use of smaller-scale surrogates with tractable likelihood functions, such as interpretability, improved communication (e.g., in our epidemic case study, we would be able to explain to a decision-maker who is familiar with the `LODE` surrogate but not with the complex ABM that \"applying intervention X in the ABM is like applying intervention Y in the `LODE`\"), and the ability to optimise probabilities/probability densities directly, thus even potentially obviating the need for any simulation from the surrogate in any case. We will discuss this more fully, using some of the additional space in the revision, and the appendix if necessary.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yc4M2fGroa", "reviewer": "Reviewer_x5Gt", "summary": "This work presents a method for learning surrogate models of simulations.\nBoth the simulation and the surrogate model are treated as causal models, where the surrogate is an element of a parametric family of causal models.\nThe map from simulation variables to surrogate variables is assumed to be known.\nFor training the surrogate, interventional consistency is used: interventions in the simulation should be well-represented by the surrogate.\nThat is, a map from simulation interventions to surrogate interventions has to be learned.\nAs a training loss, an abstraction error is used, which measures the distance between the pushforward of intervened simulation variables and intervened surrogate variables.\nThe theory section shows that zero approximation error leads to an exact transformation and that approximation errors for a given intervention are bounded.\nThe 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).", "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.\n- 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?\n- 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": "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:\n\n- How to make the distribution of the surrogate family tractable and differentiable.\n- How to select the surrogate family.\n- How to find the variable map between the simulation and the surrogate.\n\nThese 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.\n\nWhile 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?", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "- The paper is a pleasure to read: the writing is well-structured and very clear.\n- The mathematical formalism is well-defined, consistent and easy to follow.\n- Thus far, the causal abstraction literature has mostly focussed on the theoretical foundations and mathematical structure of abstractions. This paper shows an interesting application of causal abstractions and thereby bridges the gap between the causality ivory tower and useful applications.", "weaknesses": "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.\n\nReferences:\n\n[1] Kekić et al. \"Targeted Reduction of Causal Models.\" UAI (2024)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time & kind comments (\"paper is a pleasure to read..very clear\", \"well-defined, consistent and easy to follow\", \"bridges the gap between.. causality.. and useful applications\").\n\nThe reviewer identifies as a single weakness the novelty of our paper wrt a very recent UAI 2024 paper by Kekić et. al. [R3.1], presented at UAI less than 3 weeks ago. Despite this being categorized as a contemporaneous work by the NeurIPS Reviewer Guidelines, we are happy to detail the main differences between the two papers, highlight our novel contributions, & discuss this contemporaneous work in our paper. \n\nThe two papers share some similarities as they both start from an existing causal abstraction framework, adapt a measure of interventional consistency, and use it learn new models. However the two works differ substantially in aims, methodology & results:\n\n- [R3.1] has an _explanatory focus_. They deal with _targeted reduction of causal models_ aimed to generate a model \"explaining causal influences on an observable target variable $Y$\"; they describe the target variable $Y$ as a \"property of interest\" & the abstraction to $Y$ as a \"detector\" quantifying \"the presence or magnitude of a phenomenon in the data\". Thus, reduction works around a chosen focal point ($Y$) & the only causal dynamics of interest are the ones converging on the said variable. In contrast, our work has a _simulation focus_: we deal with _learning surrogate models_ that would capture the whole causal dynamics of a system of interest at a different level of abstraction. Our work does not require the modeller to commit to any target variable: in the context of our SIRS simulation, [R3.1]'s approach would require the modeller to choose which target variable is of interest (e.g.: number of infected or number of recovered); our approach, instead, provides a simplified & accurate surrogate that simulates the dynamics of all relevant variables. To provide analogous simulation results, [R3.1]'s approach would require instantiating as many reduced models as the variables of interest\n- There are also fundamental methodological differences. Since reduction is interested in a single target variable, only constructive transformations equivalent to clustering are considered & the reduced causal model is a simple collection of nodes $Z_i$ only affecting node $Y$. This trivial structure does not encode complex causal dynamics, such as influences between variables $Z_i$; it has indeed a resemblance to ICA, as explained in their discussion. On the other hand, we consider a broader class of $\\tau$ maps and arbitrary surrogate models describing complex causal structures\n- [R3.1] introduces further simplifications (linearity of $\\tau, \\omega$, affinity of the high-level mechanisms) for the sake of identifiability analysis. Experiments are run on simple physical models, as these simplifications rarely comply with real-world systems. Not requiring them, our approach is applied to actual ABMs which, as the reviewer states, \"bridges the gap between the causality ivory tower and useful applications\"\n\nThe most striking similitude between our works is how we adapted the existing interventional consistency loss by replacing a JS distance with a KL divergence, & substituting a max operator with an expectation. But these are common simplifications in ML adopted in causal abstraction papers predating both our works. Interestingly, from a similar loss function, we derived propositions that are relevant wrt our different aim: [R3.1] proves positivity, invariance to invertible reparametrization & zero for exactness; we prove zero for exactness and upper-bound for divergence.\n\nWe will briefly discuss this contemporaneous work in the Related Work section & add in the Appendix an exhaustive comparison along the lines above. A detailed comparison will be valuable to practitioners for choosing the approach that best suits their needs, whether modelling locally the dynamics of a single variable or whether simulating an entire system at a coarser scale. \n\n**Questions & limitations**\n\nQ1: Hard interventions are a requirement of the used abstraction framework as the loss is defined between posets of interventions which arise only in such a case. Our interventions are defined on parameters controlling the dynamics of the system; this modelling choice allows us to: disentangle the dynamics of the system & the params controlling it; rely on known ODEs for the dynamics; express parsimonious (defined by a single value) & interpretable (lockdown equal to setting the param to zero) interventions. In general, our work may be expanded to deal with soft interventions [R3.2], e.g., to model realistic uncertain/\"fat-fingers\" interventions by policy-makers.\n\nQ2: Just a helpful picture! The method is general.\n\nQ3: The `\\tilde{\\theta}_t` determine the update of the hidden state `z_t` of the ODE/RNN at time `t`, and `z_t` mediates the dependency of `\\tilde{y}_t` on the `\\tilde{\\theta}_{1:t}` and `I_0` (please see Appendix E). However, we have wrongly put arrows from `\\tilde{y}_t` to `\\tilde{y}_{t+1}`; will remove in revision.\n\nL: It is incorrect that we do not specify \"how to make the distribution of the surrogate family tractable and differentiable\": we show three examples for how this can be done by combining neural networks, differentiable simulators, & standard distributions in the experimental section. We will nonetheless discuss other strategies in the revision, e.g. the use of neural generative models (e.g., normalising flows) to specify the stochasticity of surrogates, & of sample-based interventional consistency losses (e.g., MMDs) for the case of differentiable surrogates with intractable distributions.\n\nFor other limitations, please see our answer to Reviewer fR4E: specifically, on choosing interventions/finding a variable map/misspecification see Q1/L1/Q2.\n\n**Refs**\n\n[R3.2] Massidda et al. \"Causal Abstraction with Soft Interventions.\" CLeaR (2022)", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sYg79uKrXR", "reviewer": "Reviewer_3hmr", "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.", "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.\n- More analysis on the performance of interventionally trained data on observational data would be beneficial.\n- perhaps some discussion on surrogates with causal inference would be relevant, see e.g., \nWitty, 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.\n- 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": "NA", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- the paper discuss a relevant practical use case for surrogate modeling\n- the concepts are clearly described under the low page limit", "weaknesses": "- the discussion of the results could be improved\n- the test case is perhaps a bit simple", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Improving the discussion of results**\n\nWe will use some of the additional space in the revision to provide further discussion. In particular, we will:\n- state and discuss the relative computational costs of our complex simulators and our surrogates (briefly: we see that our surrogates run approximately 3-30 times faster across the experiments we consider); and\n- provide a more extensive discussion of the advantages of training surrogates for interventional consistency. For example, with respect to the epidemic case study we present, we can discuss a hypothetical scenario in which a policymaker must decide whether and when to introduce a lockdown in order to minimise the average number of infections over the simulated time horizon. Please see the Global Rebuttal for further details. (Is this the kind of discussion you would like to see? We hope this provides the discussion you are looking for on the relative performance of the interventional vs observational surrogates (as in your second question); please do let us know in the discussion period if there are specific points you would like us to discuss.)\n\n**Further experiments**\n\nTo address your point that \"the test case is perhaps a bit simple\": the reason we focus on this particular epidemic simulator is that it allowed us to clearly and transparently demonstrate that simulators can be expressed as SCMs, which is important for establishing that the theory of causal abstraction can be used to learn surrogates. To nonetheless demonstrate our framework in a further, more complex setting, we have prepared another case study in which we model an ecology of species in a predator-prey relationship and the possible effects of reintroducing an additional species into the mix. Please see the Global Rebuttal for details. We emphasise that this additional case study is indeed more complex, as requested, in that it consists of agents that move around a spatial environment and reproduce and die, and consists of stocks of natural resources that grow and are consumed over time. We hope this addresses this comment, and that it provides further evidence that our framework can be applied in various settings.\n\n**Question about alternative approaches to surrogates and RNN**\n\nWe believe the approach you suggest could in principle be taken for a limited set of surrogate families. (This is not what we do in the `LRNN` model: as described in the Appendix, we feed $\\tilde{\\boldsymbol{\\theta}}$ into the surrogate at each time step, such that intervening at time $t$ amounts to feeding in a modified $\\tilde{\\boldsymbol{\\theta}}$ at that time.) \n\nHowever, this will not work generally. For example, it would not work when the surrogate family is mechanistic, such as in the case of the `LODE` family we consider in our experiments. Thus it can't be the basis of a general framework for training interventionally consistent surrogates. In contrast, our framework is useful because it handles both cases, and is thus more general.\n\nThis approach would also have the undesirable property that an intervention at time $t$ would have a causal influence on variables \"located\" at times $<t$, which by definition should not be the case. This would mean that time has different meanings in the two models, which amongst other things will reduce interpretability.\n\nNote that interventions in our surrogates are explicitly specified on abstract causal variables via $\\omega$. Consider a pandemic sim where music venues can be closed. In our surrogates, $\\omega$ specifies how venue closure can be interpreted as a direct intervention on, say, $\\alpha$ in the `LODE` surrogate. If interventions aren't transformed via $\\omega$ you lose this interpretability. You also lose out on scalability: $\\omega$ reduces the space of possible interventions significantly.\n\n**Literature recommendations**\n\nSimilarly to us [R4.1] & [R4.2] focus on improving causal reasoning methods in complex systems with causal inference. [R4.1] investigates hierarchical data settings with latent confounders, while [R4.2]'s active learning framework performs inference without assuming access to the true SCM. Instead, it assigns a Bayesian prior over the causal model class & jointly infers a posterior over both causal models (discovery) & queries of interest (inference). Both frameworks aim to handle complex causal relations which is crucial for surrogate modelling of complex systems.\n\nAdditionally, both frameworks employ GPs to model causal relationships, aligning with our objective of building surrogates when it's hard to work directly with the underlying SCM. However, they rely on several simplifying structural assumptions in order to restrict their focus to a class of causal sufficient/identifiable models. In contrast, we allow for a wider class of SCMs, resulting in a more robust framework. Further, these papers overlook the interventional consistency between the surrogate & simulator, a topic we thoroughly examine via causal abstraction. A notable distinction is our approach for addressing the computational difficulties of large-scale simulators, especially within policymaking scenarios, through a causally consistent & computationally efficient surrogate paradigm. Our goal of developing interventionally consistent surrogates aligns with the goal of accurately estimating causal effects discussed in these papers. We will update the Related Work section of the revision, & include [R4.1] & [R4.2] as examples of surrogates for causal inference. \n\n[R4.3] is indeed relevant to our work when domain expertise is available & can be incorporated into the surrogate by enforcing monotonicity on certain features. As noted, an example could be when we identify a monotonic effect in interventions. For instance, one may enforce that longer lockdowns lead to less infections. We'll incorporate this discussion into the revision.\n\n**Refs**\n\n[R4.1] Witty et al. (2020)\n\n[R4.2] Toth et al. (2022)\n\n[R4.3] Riihimäki et al. (2010)", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for their feedback. We have responded to each of your points individually & are happy to expand on anything in the discussion period. We address some common points here.

**Additional experiment**

A common recommendation was to test our method in another experiment. We have therefore designed and run a further experiment, and provide details & results here. We'll use the extra space & appendix to describe & present the results of this extra case study, & hope this addresses `fR4E`, `fUGP`, & `3hmr`'s requests to include a further example from a different policy setting.

In this extra case study, we consider a different policy scenario: reintroducing a species into an ecology, & simulating the ensuing population dynamics. Specifically, we slightly adapted a model from [G1]: we model an environment initially consisting of `grass`, `sheep`, & `wolves`, in which `grass` grows & is eaten by `sheep`, `sheep` eat `grass` & reproduce & get eaten by `wolves`, and `wolves` eat `sheep` & reproduce. The intervention we consider entails reintroducing a third animal species -- `bears`, which eat both `sheep` and `wolves`, & 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.) This additional simulator is suitable for similar reasons to the SIRS ABM: predator-prey population dynamics models as in [G1] are easy to understand & are familiar to the social simulation community, yet are complex enough to demonstrate our method's viability.

We simulate the interactions between these 4 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` (i.e. the natural resource sustaining life) over time). As in the epidemic case study, we consider the problem of learning interventionally consistent surrogates for this complex spatiotemporal simulator, & once again examine three possible approaches for constructing surrogate families:

- a family of deterministic mechanistic models based on a discrete-time Lotka-Volterra model of population dynamics [see, e.g., G2], 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);
- an `LRNN` family, exactly mirroring the `LRNN` family considered in the epidemic case study presented already;
- and a third family considers a hybrid approach, where (as in the `LODERNN` family considered in the epidemic case study) we pass a recurrent network over the underlying Lotka-Volterra-type 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 the pdf document accompanying the Global Rebuttal, 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.

**Discussion of results & building benchmarks**

Some reviewers (namely `fR4E`, `fUGP` & `3hmr`) also suggested we expand our discussion of the results and of what benchmarks should be established for causal surrogate modelling. To this end, we will specifically discuss one further approach to benchmarking (beyond the metrics appearing in Table 1) based on the performance of causal surrogates on downstream decision-making tasks. In the context of the epidemic case study presented, we will discuss and use as a benchmark metric the question of how well the different surrogates preserve the ordering of interventions (lockdown vs. no lockdown) in terms of the degree to which they reduce the number of infections over time. The SIRS ABM predicts that any lockdown is better than no lockdown at all, and we would like for our surrogates to preserve this property (i.e. that no lockdown in the surrogate is also worse than introducing any lockdown). 

We have checked for this property in both the interventional & observational surrogates, and see that the latter often do not predict that no lockdown is the worst option in this respect, and in some cases mistakenly predict that no lockdown is the _best_ option (e.g., the observational `LRNN` predicts that no lockdown was the best option in 1 of 5 training repeats, and was not the worst option in all 5 of 5 training repeats). In contrast, none of the interventional surrogates predict that no lockdown is the best option, and only the interventional `LODE` model predicts that no lockdown is not the worst option (in only 2 out of 5 training repeats). 

We will use some of the extra space & appendix to expand on the discussion of the results in this way, and use this as an example of benchmarks that can be established & used in the future literature on interventionally consistent surrogate modelling. 

**Refs**

[G1] Wilensky and Reisman "Thinking like a wolf, a sheep, or a firefly: Learning biology through constructing and testing computational theories—an embodied modeling approach." Cognition and instruction 24.2 (2006): 171-209.

[G2] Sabo "Stochasticity, predator–prey dynamics, and trigger harvest of nonnative predators." Ecology 86.9 (2005): 2329-2343.

Meta-review: 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.
Decision: Accept (poster)