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Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation
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Proposes ACM-FD, a GP-noise multi-functional DDPM with random-mask zero-regularized denoising loss and Kronecker-structured GP sampling, acting as a single probabilistic surrogate for forward, inverse, completion, and joint simulation tasks in multi-physics systems.
Problem
Existing ML surrogates for multi-physics simulation usually handle a single prediction task, need retraining for each new task, and lack uncertainty quantification; sampling GP noise over large meshes is also cubically expensive.
Approach
Extend DDPM to function space by replacing white noise with GP noise over sampling meshes, train one denoiser with random masks that zero-regularizes predicted noise on conditioned entries so any subset of functions can be conditioned on at inference, and use a multiplicative SE kernel so the GP covariance has Kronecker structure, enabling Cholesky-free sampling via tensor-matrix products.
Key results
- ACM-FD achieves lowest relative L2 error on most of 24 prediction tasks versus FNO, GNOT, DON, Simformer (e.g., D-F a,u to f: 1.59e-02 vs FNO 2.37e-02), though FNO is better on several D-R and T-F tasks.
- Kronecker-based computation reduces per-epoch training time by about 41-44% and inference time per sample by 85-87% (0.899 s vs 6.66 s on C-D).
- Empirical coverage probability for ACM-FD is closer to nominal levels than Simformer in most of 8 tasks on C-D and D-F (e.g., C-D v,s to u at 0.9: 0.939 vs 0.695).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use arbitrarily conditioned multi-functional diffusion instead of task-specific neural operators | multi-physics prediction · Darcy Flow, f,u to a | relative L2 error: 1.88e-02 (1.66e-04) -> 1.32e-02 (2.18e-04) | Helped |
| Use arbitrarily conditioned multi-functional diffusion instead of the strongest listed baseline | multi-physics prediction · Convection Diffusion, v,s to u | relative L2 error: 5.80e-03 (1.51e-04) -> 1.60e-02 (2.15e-04) | Hurt |
| Use arbitrarily conditioned multi-functional diffusion instead of FNO | multi-physics prediction · Diffusion Reaction, f1,u1 to f2 | relative L2 error: 1.07e-02 (1.92e-04) -> 1.44e-02 (8.96e-04) | Hurt |
| Use arbitrarily conditioned multi-functional diffusion instead of FNO | multi-physics prediction · Torus Fluid, w0,w5 to w1 | relative L2 error: 1.28e-02 (2.38e-04) -> 2.73e-02 (4.78e-03) | Hurt |
| Use ACM-FD instead of MFD for unconditional generation | multi-physics generation · Darcy Flow | Equation Error: 0.0584 -> 0.0576 | Helped |
| Use ACM-FD instead of MFD for unconditional generation | multi-physics generation · Torus Fluid | Equation Error: 0.0234 -> 0.0273 | Hurt |
| Use explicitly conditional training instead of unconditional training (ablation) | ACM-FD prediction · Darcy Flow, f,u to a | relative L2 error: 1.70e-1 (3.45e-3) -> 1.32e-2 (2.18e-4) | Helped |
| Use explicitly conditional training instead of unconditional training (ablation) | ACM-FD prediction · Convection Diffusion, s,u to v | relative L2 error: 5.47e-1 (3.56e-2) -> 2.17e-2 (4.53e-4) | Helped |
| Set masking probability p to 0.2 rather than p=0.5 (ablation) | ACM-FD · Darcy Flow, f,u to a | relative L2 error: 1.32e-2 -> 2.16e-2 | Hurt |
| Set masking probability p to 0.8 rather than p=0.5 (ablation) | ACM-FD · Convection Diffusion, u to v | relative L2 error: 2.66e-2 -> 5.15e-2 | Hurt |
| Remove Kronecker product-based computation (ablation) | ACM-FD · Convection Diffusion, inference | inference time per sample: 0.899 -> 6.66 | Hurt |
| Remove Kronecker product-based computation (ablation) | ACM-FD · Darcy Flow, inference | inference time per sample: 0.975 -> 6.7 | Hurt |
Surprising findings
- FNO outperforms ACM-FD on several Torus Fluid and Diffusion Reaction prediction tasks, and on one D-R task.
- Unconditional MFD achieves slightly better equation error and MRPD than ACM-FD on the Torus Fluid system.
Limitations
- Diffusion training needs about 20K epochs versus about 1K for deterministic neural operators, so total training time is higher than baselines.
- Evaluated only on four relatively simple classical systems on regular meshes, with 1,000 training instances.
- Kronecker-structured GP sampling relies on a multiplicative kernel on tensor-product mesh locations.
Experimental setup
Datasets: Darcy Flow simulations (finite-difference generated) Convection Diffusion simulations (pdepe generated) PDEBench 2D diffusion-reaction Torus Fluid simulations (finite-difference generated)
Benchmarks: Darcy Flow Convection Diffusion Diffusion Reaction Torus Fluid
Metrics: relative L2 error equation error MRPD empirical coverage probability runtime
Baselines: FNO DON PODDON GNOT Simformer MFD beta-VAE MFD-Inpaint scipy griddata interpolation
Builds on: DDPM Fourier Neural Operator Simformer PDEBench
Closest papers by meaning
- Immiscible Diffusion: Accelerating Diffusion Training with Noise Assignment (NeurIPS 2024 · similarity 0.87)
- Enhanced Diffusion Sampling via Extrapolation with Multiple ODE Solutions (ICLR 2025 · similarity 0.87)
- Tortoise and Hare Guidance: Accelerating Diffusion Model Inference with Multirate Integration (NeurIPS 2025 · similarity 0.87)
- Physics-Informed Diffusion Models (ICLR 2025 · similarity 0.87)
- Provable Posterior Sampling with Denoising Oracles via Tilted Transport (NeurIPS 2024 · similarity 0.87)
- EVODiff: Entropy-aware Variance Optimized Diffusion Inference (NeurIPS 2025 · similarity 0.87)
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