Sim+Real: Joint Simulation - Experiment Training Improves Balanced Prediction in Physical Systems

1 Introduction

• Simulations provide dense trajectories and potentially inaccessible variables, whereas experiments directly observe the physical system but are costly, noisy, and often partial.Combining them means balancing abundant but imperfect numerical supervision with limited measurements.Source: §1 · p. 1
• The paper asks how one model can remain useful on both domains while balancing experimental adaptation against simulation forgetting.It distinguishes the pre-existing simulation-to-experiment distribution gap from increased held-out simulation error after experimental fine-tuning.Source: §1 · p. 1
• Joint training keeps simulation and experimental supervision active, retaining a simulation-gradient contribution while adapting to experimental data.The paper formulates prediction as a multi-objective problem with domain-specific risks; conflicting gradients can let experiment-only updates increase simulation error.Source: §1 · p. 2
• Because simulations and experiments may contain different fields and grids, the study uses MORPH, which supports the heterogeneous inputs without discarding simulation variables or padding unobserved experimental fields.The architecture and datasets provide a controlled setting for testing the training-protocol hypothesis.Source: §1 · p. 2
• The study compares Simulation only, Experiment only, Sim→Exp, and Joint training with matched initialization and optimization settings across four fluid systems and two capacities.Every model is tested on both domains, and the claim concerns balanced dual-domain performance rather than universal superiority on either domain.Source: §1 · p. 2

2 Experimental Setup

• MORPH processes the different field sets and, in some systems, spatial grids without padding missing experimental fields or discarding additional simulation fields.The study trains the models from scratch and holds the architecture fixed across training protocols.Source: §2 · pp. 2–3
• The four RealPDEBench fluid systems share measured velocity fields (u, v), while simulations additionally contain pressure; the table reports native grids.Cylinder, FSI, and Foil use 20-frame input and target blocks; Controlled Cylinder uses 10-frame blocks.Source: §2 · p. 2, Table 1 · p. 3
• MORPH predicts one future frame from a fixed-length teacher-forced context, then rolls predictions forward autoregressively through the target horizon at test time.This frame-wise protocol exposes error accumulation and long-horizon stability, and differs from RealPDEBench baselines that predict the full target block in one round.Source: §2 · p. 3
• For each dataset and capacity, all four training modes start from the same initialization and use the same within-dataset batch size.The two model capacities are 17.5M and 48.2M trainable parameters; Joint uses equal source weights.Source: §2 · p. 3
• Every checkpoint is evaluated on held-out simulation and experimental data using shared velocity fields, with equal-domain RMSE, relative L2, and fRMSE summaries.The summaries weight the two domains equally, independently of test-set size.Source: §2 · p. 3

3.1 Joint training gives the best balanced dual-domain performance.

• Joint is best in 24/24 balanced comparisons; against the best non-joint protocol, mean improvements are 49.1% bRMSE, 52.4% bRelL2, and 53.0% bfRMSE.The metrics cover physical-space, relative, and Fourier-space error.Source: §3.1 · pp. 3–4, Table 2 · p. 4
• Joint is best in all 24 dataset–capacity–metric comparisons for evaluation weights λ ∈ {0.2, 0.3, . . . , 0.8}, and in 22/24 at λ = 0.1 and 21/24 at λ = 0.9.The trained checkpoints are fixed while the evaluation weights vary.Source: §3.1 · p. 4

3.2 Sequential fine-tuning specializes to experiment and forgets simulation.

• Sim→Exp raises simulation RMSE over Simulation only in all eight settings; Joint reduces Sim→Exp simulation RMSE by 60.0–87.2% (median 66.9%).The results provide test-set evidence of simulation-domain forgetting under sequential adaptation and better retention under Joint.Source: §3.2 · p. 4
• Sim→Exp has the lowest experimental RMSE in six of eight settings, while Joint is best in the two Cylinder settings.Sequential fine-tuning can specialize more aggressively to experiment; the balanced result does not imply universal experimental-domain dominance.Source: §3.2 · p. 4
• Simulation only has the lowest simulation RMSE in six of eight settings, while Joint is best for Cylinder-S and FSI-S; experimental winners also vary by dataset and capacity.Joint has the lowest experimental RMSE for both Cylinder capacities, while Sim→Exp wins for both capacities of Controlled Cylinder, FSI, and Foil.Source: §3.2 · p. 4
• For pressure, which is supervised in simulation but absent from experimental measurements, Joint reduces RMSE by 52.4–86.6% relative to Sim→Exp.This indicates better preservation of this simulation-only field during experimental adaptation.Source: §3.2 · p. 4

4 Conclusion

• Joint achieves the best equal-domain balanced RMSE, relative L2, and Fourier RMSE in all eight settings, while Sim→Exp raises simulation RMSE in every setting.Sequential fine-tuning nevertheless gives the lowest experimental RMSE in six of eight settings.Source: §4 · p. 4
• When domain gradients conflict, experimental updates can increase simulation risk; Joint retains a simulation-gradient term and continues optimizing both risks.The conclusion offers this gradient analysis as an explanation for Joint's retention advantage.Source: §4 · p. 4

A Related Work

• The paper does not introduce a new architecture; it studies training protocols for models learning from numerical and experimental data.The related-work discussion situates the question among models for physical dynamics and PDE foundation models.Source: A Related Work · p. 7
• Unlike conventional simulation-to-real transfer, the paper asks how experimental adaptation affects simulation performance when the final model must remain useful on both domains.The related-work discussion notes that simulation data remain available during adaptation.Source: A Related Work · p. 7

B.1 Problem Setup

• The setup assigns each domain its own data distribution, training set, and risk, while the primary dual-domain evaluation uses the common measured fields (u, v).For the four fluid systems, training uses all supervised dynamic fields available in each domain.Source: §B.1 · p. 8

B.2 Proposed Modeling Requirement for Heterogeneous Data

• Field-wise tokens are formed on each domain's native grid, then a fixed latent query fuses the variable collection; unavailable experimental fields are omitted.This provides the prediction backbone a fixed latent representation despite changes in observed field identity or count.Source: §B.2 · p. 8

B.3 Sequential and Joint Objectives

• Joint training scalarizes simulation and experimental risks, using one mini-batch from each domain to estimate the weighted objective's gradient.The Joint gradient is an unbiased estimator under independent mini-batch sampling.Source: §B.3 · p. 9

B.4 Why sequential fine-tuning can forget

• For a small step, conflicting simulation and experimental gradients can make an experimental-risk-reducing update increase simulation risk.Sequential fine-tuning has no simulation-descent term once its experimental stage begins; Joint retains both objectives.Source: §B.4 · p. 9

C Experimental details

• The final configurations use p4_b6_10k for Cylinder, Controlled Cylinder, and FSI, and p4_b3_10k for Foil.These settings document the final experiment code and configurations for reproducibility.Source: C Experimental details · p. 10

C.1 Dataset construction and preprocessing

• Public benchmark splits are used without test samples in optimization or checkpoint selection; numerical-training augmentation is limited to that loader, and all protocols reuse normalization statistics from clean numerical training data.Validation, test, experimental training, and normalization data are unaugmented.Source: §C.1 · p. 10, Table 3 · p. 10

C.2 Field mapping and heterogeneous inputs

• Numerical samples use dynamic fields (u, v, p), while experiments use (u, v); Controlled Cylinder also adds two static condition maps.Unavailable experimental observables are not passed as placeholder fields.Source: §C.2 · p. 10

C.3 Forecast target construction and autoregressive evaluation

• Training forms a one-step teacher-forced pair, but validation and test autoregressively append each prediction until the benchmark target block is generated.Controlled Cylinder uses 10 context steps; the other datasets use 20.Source: §C.3 · p. 11

C.4 Architecture requirements and selection.

• MORPH is chosen as the fixed enabling architecture because its field-wise interface handles variable grids and fields without padding missing experimental fields or discarding simulation fields.The architecture comparison concerns interface capabilities, not predictive accuracy.Source: §C.4 · p. 11, Table 4 · p. 11

C.5 MORPH Architecture and Prior Applications

• MORPH combines component-wise convolutions, inter-field cross-attention, and factorized axial attention, with UPTF-7 providing a common representation across heterogeneous datasets.The paper describes prior adaptations for inverse parameter estimation and out-of-distribution transfer.Source: §C.5 · p. 11

C.6 Architecture instantiation

• The two instantiated MORPH capacities use shared settings including patch size 4, 32 field-fusion cross-attention heads, and no released pretrained weights.The configurations also use at most 20 fields and 2,048 patches.Source: §C.6 · pp. 11–12, Table 5 · p. 12

C.7 Source-exposure-matched comparison

• Joint and Sim→Exp each receive U source-specific mini-batches and UBd sample presentations from each source, but Joint uses U optimizer steps versus 2U for Sim→Exp.That is 60,000 presentations per source for Cylinder, Controlled Cylinder, and FSI, and 30,000 for Foil.Source: §C.7 · p. 12, Table 7 · p. 13

C.8 Optimization and protocol budgets

• Joint applies 0.5Lsim and 0.5Lexp gradients from separate mini-batches in one optimizer step; Sim→Exp resets optimization state before its experimental stage.For Sim→Exp, only learned network weights transfer from the best simulation-stage checkpoint.Source: §C.8 · pp. 12–13

C.9 Validation, checkpoint selection, and randomness

• Checkpoints are selected by experimental-validation RMSE at four stage updates, and the numerical test split is never used for model selection.The same selection rule applies to each protocol and both Sim→Exp stages; it favors the experimental side.Source: §C.9 · p. 13

C.10 Test metrics and common-field evaluation

• Final models are evaluated separately on experimental and numerical test loaders, with primary dual-domain scores restricted to shared velocity fields (u, v).Numerical pressure is excluded from the primary score and evaluated separately as a missing-field retention test.Source: §C.10 · p. 13

D Extended Results

• Table 8 reports full dual-domain test performance across four fluid datasets.Results are seed 0.Source: Table 8 · p. 14
• Simulation and experimental metrics are computed on shared velocity fields (u, v), with lower values indicating better performance.Source: Table 8 · p. 14
• Bold marks the best protocol for each model capacity, test domain, and metric.Source: Table 8 · p. 14

E Sensitivity to Simulation–Experiment Evaluation Weighting.

• Joint has the lowest error in all 24 dataset–capacity–metric comparisons for λ=0.2–0.8, and in 22/24 at λ=0.1 and 21/24 at λ=0.9.The checkpoints remain fixed while the evaluation weighting varies.Source: E Sensitivity to Simulation–Experiment… · p. 15
• Joint remains best across λ=0.1–0.9 for Cylinder and Controlled Cylinder, λ=0.1–0.9 for Foil MORPH-Ti, and λ=0.1–0.8 for Foil MORPH-S.For FSI, Joint is best across λ=0.2–0.9 for both capacities.Source: Figure 1 · p. 15, Figure 2 · p. 16, Figure 3 · p. 16
• At the single-domain extremes, Sim→Exp or Simulation only sometimes leads, with endpoint winners varying by dataset, capacity, and metric.Joint retains some endpoint wins, including MORPH-S for Cylinder and Controlled Cylinder at λ=1 and FSI at λ=1.Source: Figure 1 · p. 15, Figure 2 · p. 16, Figure 3 · p. 16

F Missing-Field Retention During Experimental Adaptation

• Pressure is supervised in numerical data but absent from experimental measurements, so Sim→Exp receives no direct pressure supervision during its experimental stage.Joint continues to receive a numerical mini-batch at every update and retains direct pressure supervision while adapting.Source: F Missing-Field Retention During… · p. 18
• Joint reduces pressure RMSE by 52.4–86.6% (median 61.6%), relative L2 by 54.1–87.0% (median 60.5%), and fRMSE by 58.8–86.9% (median 64.8%) versus Sim→Exp.Joint is better in all 24 pressure-metric comparisons.Source: F Missing-Field Retention During… · p. 18, Table 9 · p. 18
• The pressure comparison covers Sim→Exp and Joint; Experiment only is omitted because that protocol never supervises pressure.The consistent advantage spans physical-space RMSE, relative L2, and Fourier-space RMSE.Source: Table 9 · p. 18, F Missing-Field Retention During… · p. 18

• Training-time sensitivity to λ remains unexplored because Joint is trained only with λ = 0.5.Source: G Limitations · p. 18
• The gradient analysis is local, precedes Adam preconditioning, and does not directly measure gradient alignment.Source: G Limitations · p. 18
• Absolute errors are not an architecture-matched reproduction of RealPDEBench baselines because training uses one-step teacher forcing and evaluation is autoregressive.Source: G Limitations · p. 18