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M2PDE: Compositional Generative Multiphysics and Multi-component PDE Simulation
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Proposes M2PDE, a compositional diffusion-model framework that learns conditional energy functions per physical process or component, enabling coupled multiphysics solutions from decoupled training data and large multi-component structures from small-structure training.
Problem
Multiphysics and multi-component PDE simulation require integrating multiple specialized solvers or simulating large structures, which is costly to develop and compute; existing ML surrogates need coupled training data and fail to extrapolate to larger structures.
Approach
Model the joint solution of coupled processes/components as a product of learned conditional energy functions (via diffusion denoisers conditioned on other fields/neighbors), trained on decoupled or small-structure data, and sample jointly at inference with an iterative external loop (EM-like) over conditional diffusion steps.
Key results
- On the prismatic fuel element task, M2PDE + Transolver reaches temperature error 0.793e-2 and strain error 2.07e-2 on the 64-component structure, while the surrogate + Transolver gets 1.25e-2 and 3.31e-2; surrogate + Geo-FNO diverges.
- On nuclear thermal coupling with U-Net, M2PDE reduces coupled-solution relative L2 error for neutron, solid, and fluid fields to 1.38, 2.75, and 3.86 (x1e-2), versus 4.45, 18.2, and 8.03 for the surrogate.
- The surrogate model is more accurate on single-process decoupled predictions, and M2PDE is reported up to 41-fold faster sampling with DDIM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use M2PDE instead of the surrogate for coupled u prediction | FNO on reaction-diffusion · Reaction-diffusion, coupled u | relative L2 error: 0.0600 -> 0.0290 | Helped |
| Use M2PDE instead of the surrogate for coupled v prediction | FNO on reaction-diffusion · Reaction-diffusion, coupled v | relative L2 error: 0.0320 -> 0.0264 | Helped |
| Use M2PDE instead of the surrogate for coupled u prediction | U-Net on reaction-diffusion · Reaction-diffusion, coupled u | relative L2 error: 0.0184 -> 0.0141 | Helped |
| Use M2PDE instead of the surrogate for coupled v prediction | U-Net on reaction-diffusion · Reaction-diffusion, coupled v | relative L2 error: 0.0174 -> 0.0174 | No clear effect |
| Use M2PDE instead of the surrogate | FNO on nuclear thermal coupling · Nuclear thermal coupling, coupled neutron field | relative L2 error: 22.1 -> 8.42 | Helped |
| Use M2PDE instead of the surrogate | FNO on nuclear thermal coupling · Nuclear thermal coupling, coupled solid temperature | relative L2 error: 31.8 -> 9.72 | Helped |
| Use M2PDE instead of the surrogate | FNO on nuclear thermal coupling · Nuclear thermal coupling, coupled fluid field | relative L2 error: 10.2 -> 7.31 | Helped |
| Use M2PDE instead of the surrogate | U-Net on nuclear thermal coupling · Nuclear thermal coupling, coupled neutron field | relative L2 error: 4.45 -> 1.38 | Helped |
| Use M2PDE instead of the surrogate | U-Net on nuclear thermal coupling · Nuclear thermal coupling, coupled solid temperature | relative L2 error: 18.2 -> 2.75 | Helped |
| Use M2PDE instead of the surrogate | U-Net on nuclear thermal coupling · Nuclear thermal coupling, coupled fluid field | relative L2 error: 8.03 -> 3.86 | Helped |
| Use M2PDE instead of the surrogate | Geo-FNO on prismatic fuel elements · Prismatic fuel element, 64-component temperature | relative L2 error: divergent -> 0.950 | Helped |
| Use M2PDE instead of the surrogate | Transolver on prismatic fuel elements · Prismatic fuel element, 64-component temperature | relative L2 error: 1.25 -> 0.793 | Helped |
Surprising findings
- On the reaction-diffusion task, M2PDE with FNO has a notably larger decoupled error for v (0.0102 vs 0.0080 for the surrogate, and 0.0264 vs 0.0320 coupled), so the advantage is not uniform.
- Surrogate models fail on coupled nuclear thermal solutions (e.g., solid temperature error 31.8e-2 with FNO) even though they are accurate on decoupled data.
Limitations
- Prediction errors on coupled solutions remain higher than single-process predictions and models trained on coupled data.
- Experiments use simplified engineering problems (1D reaction-diffusion, pin-cell, plane-strain steady state) rather than full real-world systems.
- Sampling remains iterative and relatively costly; faster sampling algorithms are left to future work.
Experimental setup
Datasets: reaction-diffusion simulations (solve_ivp, FitzHugh-Nagumo) nuclear thermal coupling data (MOOSE) prismatic fuel element data (MOOSE)
Benchmarks: FitzHugh-Nagumo reaction-diffusion nuclear thermal coupling prismatic fuel element (16 and 64 components)
Metrics: relative L2 error
Baselines: surrogate model (per-field FNO/U-Net with iterative combination) GIN SAN MeshGraphNet CoAE-MLSim-style surrogate
Builds on: denoising diffusion probabilistic models product of experts Fourier neural operator Transolver Geo-FNO
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