Counterfactual Predictions in Scientific Emulators Without Controlled Experiments

1 INTRODUCTION

• When correlated drivers are not independently varied, a flexible model can fit observed data by using one driver as a proxy, yet fail when a what-if query breaks those dependencies.Controlled experiments can supply the missing variation but may be costly, require a simulator, and inherit its assumptions; some systems lack such data.Source: §1 · pp. 1–2
• ReRoute fixes the targeted input in the pretrained backbone at a reference value and routes its variation through a known pathway, leaving downstream responses to be learned from factual data.The same factual data are used for fine-tuning and the pathway receives the requested value at query time.Source: §1 · p. 2, Figure 1 · p. 3
• Under explicit structural assumptions, the target interventional response is identifiable from observational trajectories together with the known mechanism.The paper reports that the core argument is machine-checked in Lean.Source: §1 · p. 2
• ReRoute applies when a pretrained emulator has an entangled targeted input and domain knowledge can specify how that input first enters the system, without describing all downstream effects.Examples include climate forcings, catalyst concentration, exercise intensity, and nutritional composition.Source: §1 · p. 2

Summary of Results

• 18.2–31.8% lower aggregate error under severe SST–CO2 shifts, with comparable skill under standard and mild shifts, and up to 89.6% lower counterfactual response error in advection–diffusion.The ERA5-trained emulator also preserves substantially more near-surface warming under fixed CO2 and yields a tropical-cyclone climatology closer to ERA5.Source: §1 · p. 3

Scientific generalization beyond observed regimes

• ReRoute uses a query-specific assumption about how one targeted variable enters the system, rather than requiring representations to transfer across arbitrary shifts or sufficient observational support for the target regime.The cited approaches include structural invariance and observational-support methods [Yao et al., 2025].Source: §2 · p. 3

Hybrid scientific modeling

• Unlike hybrid models that encode governing equations, conservation relations, or substantial dynamics, ReRoute requires only the targeted variable’s entry pathway and leaves downstream interactions to the pretrained emulator.Examples of hybrid-model approaches include physics-informed objectives and learned dynamics combined with known mechanisms [Pan & Duraisamy, 2018; Rackauckas et al., 2020].Source: §2 · p. 3

Climate emulators and forcing disentanglement

• For ACE2’s entangled SST–CO2 inputs, ReRoute supplies the known radiative entry pathway while retaining learned dynamics, complementing controlled random-CO2 simulations that independently vary the forcings.ACE2-SOM extends the distribution to different equilibrium CO2 climates, while Clark et al. (2026) use random-CO2 simulations to address SST–CO2 entanglement.Source: §2 · p. 4

3.1 PROBLEM SETUP: TARGETED WHAT-IF QUERIES

• A targeted what-if query predicts the response at a requested driver value while holding the other observed conditions fixed; the causal analysis treats this as an intervention on that driver.The climate example asks what would happen if CO2 stayed at its 1940 level while other boundary conditions evolved historically.Source: §3.1 · p. 4

3.2 A KNOWN PATHWAY FOR THE TARGETED DRIVER

• The known pathway Γ(s0, s) encodes how changing the targeted driver from reference s0 directly enters the response, without needing to specify full system dynamics.In the climate application, it describes CO2’s immediate atmospheric effect while later climate adjustment remains learned.Source: §3.2 · p. 4

3.3 REFERENCE-CONDITIONED FINE-TUNING

• ReRoute initializes from a pretrained emulator, fixes the backbone’s targeted input at s0, and routes factual variation through Γ while fine-tuning against the original training targets.The backbone’s constant reference input prevents the targeted driver’s variation from entering its prediction through another route; fine-tuning adjusts to this division of responsibility.Source: §3.3 · p. 5

3.4 IDENTIFIABILITY OF TARGETED INTERVENTIONS

• The target intervention is identifiable when the known pathway fully mediates the targeted variable’s direct effect, observed pathway variation identifies the downstream response, and factual responses determine the response at the requested pathway value.The paper distinguishes setting S directly to s′ from conditioning on cases where S naturally equals s′.Source: §3.4 · p. 5
• The known pathway supplies how the targeted variable enters the system, while factual observations teach the downstream response that is evaluated at the pathway value for the requested intervention.This is the paper’s account of how observational trajectories can support what-if predictions without explicitly modeling downstream mechanisms.Source: §3.4 · p. 5

3.5 ReRoute FOR AUTOREGRESSIVE SYSTEMS

• In autoregressive systems, the model routes each timestep’s targeted forcing through the pathway, feeds its predicted state into the next step, and lets learned dynamics propagate subsequent effects.For a query, only the pathway input changes to the requested forcing value.Source: §3.5 · p. 6

4.1 ADVECTION–DIFFUSION TRANSPORT

• In the controlled advection–diffusion system, joint rerouting of correlated controls (c, D) performs best across most of the parameter grid; rerouting D alone also reduces out-of-distribution error.For a representative intervention after 100 rollout steps, RMSE falls from 0.1182 to 0.0108 when rerouting D and to 0.0079 when jointly rerouting (c, D).Source: §4.1 · p. 6
• Across 100-step rollouts, joint rerouting reduces step-100 average error to 0.0082 from the baseline 0.0784, an 89.6% reduction.Fine-tuning uses 10-step rollouts; the evaluation extends to 100 steps. Rerouting D alone yields 0.0130 (83.4% reduction).Source: §4.1 · p. 7, Figure 2 · p. 7

4.2.1 COUPLED CLIMATE WITH HELD-OUT COUNTERFACTUAL EVALUATION

• 18.2% under abrupt-4×CO2 and 31.8% under held-out random-CO2: ReRoute reduces aggregate climate error while remaining comparable under standard AMIP and mild fixed-CO2 forcing.The random-CO2 error falls from 0.1135 to 0.0775; abrupt-4×CO2 uses 36 initial-condition members.Source: Figure 3 · p. 8, §4.2.1 · p. 8

4.2.2 REAL-WORLD CLIMATE REANALYSIS

• Under fixed CO2 in ACE2-ERA5, ReRoute substantially corrects widespread baseline cooling and preserves more surface warming over most of the globe.The qualitative test holds CO2 at its 1940 value while retaining historical boundary conditions such as SST and sea ice; no real-world counterfactual reference is available.Source: §4.2.2 · pp. 8–9, Figure 4 · p. 9

5 DISCUSSION

• ReRoute fixes the queried input of a pretrained backbone at a reference value, then reintroduces its factual value through a known pathway while learning downstream responses from observations.The framework requires neither controlled intervention data nor a complete mechanistic model.Source: §5 · p. 9
• 18.2–31.8%: ReRoute reduces aggregate error under severe SST–CO2 shifts, while preserving skill under standard and mild shifts.Across controlled and realistic settings, the largest gains occur under the strongest shifts; the ERA5-trained emulator also restores substantially more near-surface warming under fixed CO2.Source: §5 · p. 9
• The theory covers one targeted variable with a state-independent pathway; imperfect pathway knowledge may limit effectiveness.Experiments indicate effectiveness can persist with state-dependent pathways and multiple jointly rerouted variables.Source: §5 · p. 9

ETHICS STATEMENT

• ReRoute is intended for scientific research, not as a substitute for established physical climate models or direct operational or policy guidance.Predictions may inherit limitations from the data, pretrained emulator, and supplied pathway, especially under extreme distribution shifts.Source: ETHICS STATEMENT · p. 10

A.2 SOURCE-INTERFACE REDUCTION

• The causal reduction identifies an intervention by evaluating the learned downstream response at the pathway value corresponding to the target input.The intervention fixes the known pathway at its target value, reducing the causal problem to identifying that downstream response.Source: §A.1 · p. 16, §A.2 · p. 18
• The source-interface response captures the known direct contribution of the targeted input, while the remaining downstream mechanisms need not be specified.The downstream response map describes how the system reacts to changes in this interface.Source: §A.2 · pp. 16, 18

A.3 OBSERVATIONAL IDENTIFICATION OF THE DOWNSTREAM RESPONSE

• Under source exogeneity, observational variation in the pathway identifies the downstream response map only up to null sets of the observed pathway distribution.The response at a particular target value is not fixed by this almost-everywhere information alone.Source: §A.3 · pp. 19–20
• When the target pathway value has zero observational probability, identifying its response requires an additional target-determination condition.This issue can arise even when the target lies within the observed pathway support.Source: §A.3 · p. 20

A.4 ADMISSIBLE MODELS, TARGET DETERMINATION, AND IDENTIFICATION

• If target determination holds, any two admissible models with the same observational distribution induce the same target interventional distribution.If the condition holds for every realizable observational distribution, the target is identifiable within the admissible model class.Source: §A.4 · pp. 20–21
• The target-determination condition upgrades agreement of response maps almost everywhere to agreement at the particular target pathway value.The two models share the observed pathway law, so the condition is evaluated at that common law.Source: §A.4 · pp. 20–21
• For a positive-mass target, target determination holds for every response-map class; for an in-support, zero-mass target, continuity is sufficient.For an out-of-support target, target determination is an extrapolation assumption that observational data cannot verify; it must follow from the model class or additional mechanistic structure.Source: §A.4 · p. 21

A.5 LEAN VERIFICATION

• The core identification argument was machine-checked in Lean 4 across roughly 900 lines and four modules.The development covers intervention-to-pathway reduction, observational recovery under source exogeneity, and the final target-identification theorem.Source: §A.5 · p. 22

B ADVECTION–DIFFUSION EXPERIMENT

• The section provides additional details about the controlled advection–diffusion experiments.Source: B ADVECTION–DIFFUSION EXPERIMENT · p. 23

B.1 DATASET AND PRETRAINED FNO BACKBONE

• The experiment tests ReRoute against known ground truth in a two-dimensional advection–diffusion system with state-dependent pathways and one or two targeted controls.Both the exact dynamics and pathway are available in this setting.Source: §B.1 · p. 23
• Training trajectories contain strongly correlated advection speed and diffusivity, with parameter correlation r = 0.989.The dataset has 512 unique training trajectories across 32 initial conditions; validation and test sets each use eight independent initial conditions.Source: §B.1 · p. 23
• The baseline is a pretrained Fourier neural operator evaluated over 100-step rollouts, ten times its 10-step factual training horizon.On 128 held-out in-distribution trajectories, its factual RMSE is 0.00184 after 10 steps and 0.01448 after 100 steps.Source: §B.1 · p. 23

B.2 STATE-DEPENDENT PATHWAY

• The exact pathway computes the state-dependent change from reference parameters at every rollout step, rerouting either diffusivity alone or both advection speed and diffusivity.Because the exact update is available, the pathway introduces no approximation error, isolating whether fine-tuning uses it effectively.Source: §B.2 · p. 24

B.3 EVALUATION AND ADDITIONAL RESULTS

• Jointly rerouting (c, D) gives the lowest counterfactual-response error across most of the held-out grid, while rerouting D alone also improves on the baseline.The evaluation covers a 41 × 41 grid, including combinations far from the strongly correlated training support.Source: §B.3 · p. 24, Figure 6 · p. 25
• Both ReRoute variants maintain substantially lower error through 100-step rollouts, with jointly rerouting (c, D) slowing error growth beyond the ten-step fine-tuning horizon.The evaluation grid was held out from training and fine-tuning.Source: §B.3 · p. 24
• The formal guarantees do not directly cover state-dependent pathways or jointly rerouted controls, although both extensions work in these experiments.Joint rerouting performs best because c translates spatial structure while D damps smaller scales; rerouting only D leaves the backbone to extrapolate the advection response.Source: §B.3 · p. 24
• At c = 0.5 and D = 0.018, 100-step RMSE falls from 0.1182 for the baseline to 0.0108 with D rerouting and 0.0079 with joint rerouting.Both ReRoute variants reduce spatially coherent field-level errors rather than relying on cancellation across space.Source: §B.3 · p. 24, Figure 7 · p. 25

C CLIMATE DATA AND PRETRAINED MODELS

• The climate experiments use ACE2-family emulators, including an ACE2S-SHiELD+ model trained without random-CO2 data and an ACE2-ERA5 model trained on historical reanalysis.Both models evolve the atmosphere autoregressively at six-hour resolution on a 1° Gaussian grid with eight atmospheric layers.Source: C CLIMATE DATA AND PRETRAINED MODELS · p. 26

C.1 ACE2S-SHIELD+no-RC: CONTROLLED CLIMATE TRAINING DATA

• The controlled-climate experiments use the released ACE2S-SHiELD+no-RC checkpoint, whose underlying SHiELD simulations are regridded and vertically coarsened before training.The source simulations use C96 resolution and 79 hybrid sigma–pressure levels; the model data use a 1° Gaussian grid and eight atmospheric layers.Source: §C.1 · p. 26
• The no-RC checkpoint's 90-year training set combines 45 years of AMIP simulations with 45 years of equilibrium-climate simulations.The equilibrium data span 1×, 2×, and 4× reference CO2 concentrations.Source: §C.1 · p. 26
• The checkpoint receives no training trajectories in which CO2 varies independently of the corresponding climate state.AMIP uses prescribed historical SST, sea ice, and CO2; SST is also prescribed when equilibrium data are used for ACE2S training.Source: §C.1 · p. 26
• ACE2S-SHiELD+ differs from ACE2-ERA5 by additionally predicting total frozen precipitation rate and by receiving CO2 as a next-timestep forcing.The specified CO2 concentration corresponds to the six-hour interval of the state update and associated boundary-energy fluxes.Source: §C.1 · p. 26

C.2 ACE2-ERA5: HISTORICAL REANALYSIS TRAINING DATA

• ACE2-ERA5 uses processed ERA5 data spanning 1940–2022, with 1940–1995, 2011–2019, and 2021–2022 for training and 1996–2000 for validation.The held-out test periods are 2001–2010 and 2020.Source: §C.2 · p. 26
• ERA5 fields are represented on a 1° Gaussian grid with eight hybrid sigma–pressure layers at six-hour resolution.State variables are instantaneous snapshots, whereas radiative, turbulent, and precipitation fluxes are six-hour means.Source: §C.2 · p. 27
• The ACE2-ERA5 interface distinguishes prognostic inputs and outputs, input-only forcing variables, and diagnostic outputs that are not fed back autoregressively.Surface skin temperature is prognostic over land and sea ice, but prescribed as SST over open ocean.Source: §C.2 · p. 27, Table 2 · p. 27

D COMPUTATIONAL DETAILS

• ReRoute fine-tuning runs took less than 24 hours on two NVIDIA A100 GPUs, and three-seed runs showed no material convergence or downstream-performance differences.Uncertainty is reported across stochastic realizations and initial conditions, not fine-tuning seeds, because seed variation was small relative to predictive variability.Source: D COMPUTATIONAL DETAILS · p. 28
• The reported 18× compute reduction compares ReRoute adaptation with training a controlled-data ACE2S-SHiELD+ counterpart on the controlled SHiELD benchmark only.The comparison excludes backbone training and dataset or evaluation-simulation generation costs, and is not made for ACE2-ERA5.Source: D COMPUTATIONAL DETAILS · p. 28

E.1 RRTMG CO2 PATHWAY CONSTRUCTION

• The CO2 pathway uses RRTMG responses at a fixed climatological-mean atmospheric state to isolate the instantaneous radiative effect of changing CO2.Stored response coefficients are scaled by log(st/s0); subsequent atmospheric and slab-ocean changes emerge through autoregressive ACE2-SOM dynamics.Source: §E.1 · pp. 28–29

E.2 TROPICAL CYCLONE TRACKING DETAILS

• Tropical cyclones are detected and tracked with the same TempestExtremes v2.1 pipeline across ERA5, the pretrained ACE2-ERA5 baseline, and ACE2-ERA5+ReRoute.Using the common tracker permits direct comparison of model-generated cyclone statistics against ERA5.Source: §E.2 · p. 29

F.1 COUPLED CLIMATE SIMULATION

• The abrupt-4×CO2 experiment starts from a 1×CO2 climate state, applies 4×CO2 forcing, and evaluates 36 ensemble members over 90-day rollouts against corresponding SHiELD references.This forcing–state combination is absent from training; the backbone remains conditioned on the reference CO2 while the intervention enters through the fixed RRTMG pathway.Source: §F.1 · p. 30
• The member-wise aggregate error is 0.1324 ± 0.0101 with ReRoute versus 0.1619 ± 0.0116 for the native no-RC model, an 18.2% reduction in mean member-wise error.Member-wise error exposes variability across independent initial conditions, complementing the ensemble-mean-first metric.Source: §F.1 · p. 30

• ACE2-ERA5 is a six-hour autoregressive global emulator trained directly on ERA5, where historical co-evolution of SST, sea ice, and CO2 provides no independently controlled variations.Source: §F.2 · p. 31
• ReRoute starts from the same released checkpoint, adapts only with factual historical ERA5 samples, and changes only the CO2 pathway while leaving the architecture and other emulator components unchanged.Source: §F.2 · p. 31
• Two 1940–2020 rollouts differ only in CO2 trajectory; with SST and sea ice held identical, their time-mean 2-m temperature difference isolates the changed CO2 pathway, without an observed counterfactual reference.Source: §F.2 · p. 31
• ReRoute produces a much weaker fixed-minus-historical near-surface temperature response than ACE2-ERA5, preserving substantially more warming encoded by prescribed SST and sea-ice evolution.Source: §F.2 · p. 31
• With identical cyclone detection and tracking across ERA5 and both rollouts, ReRoute’s held-out 2001–2010 track distribution more closely matches ERA5 than the pretrained baseline.Source: Figure 8 · p. 32, §F.2 · p. 32