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Ultra Fast PDE Solving via Physics Guided Few-step Diffusion
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Phys-Instruct distills a multi-step EDM diffusion teacher into a one- to four-step PDE generator, adding a differentiable PDE residual penalty during distillation so no physics guidance is needed at inference.
Problem
Diffusion-based PDE solvers need hundreds of denoising steps and often produce samples with large physics (PDE) residuals; existing physics-guided methods add cost at inference or trade quality for physical fidelity.
Approach
Distill a frozen multi-step diffusion teacher into a few-step student via integral KL distillation, with a differentiable mean-squared PDE residual penalty applied to student samples during training only, so inference is a fixed few-step sampler with no physics guidance.
Key results
- On Darcy flow, one-step Phys-Instruct reaches PDE error 3.15e-1 versus 5.59e-1 for 200-step EDM and 4.38e-1 for 200-step DiffusionPDE.
- Four-step Phys-Instruct attains the lowest PDE error on all five benchmarks in the unconditional table (e.g., Burgers 5.54e-2 vs 3.84e-1 for 200-step CoCoGen).
- On Helmholtz forward, one-step Phys-Instruct achieves 1.86% relative error versus 9.69% to 13.76% for 2000-step baselines.
Surprising findings
- Physics guidance without the penalty yields PDE error 8.75 on Darcy flow, versus 3.15e-1 with it, a roughly 27x reduction.
- The 200-step diffusion baselines with physics guidance at sampling time are sometimes worse than the 4-step Phys-Instruct student, so aggressive step reduction degrades baselines more than the distilled student.
Limitations
- Physics penalty weight lambda_phys is sensitive: too small is ineffective and too large (5e-2) diverges.
- Evaluated only on 2D or 1D benchmarks at modest resolutions (up to 128x128); higher-dimensional PDEs are left to future work.
- Burgers and NS residuals are simplified or use finite differences rather than full time-dependent PDE error at the NS terminal state.
Experimental setup
Datasets: Darcy flow Poisson non-bounded Navier-Stokes Burgers' equation inhomogeneous Helmholtz
Benchmarks: Darcy flow Poisson non-bounded Navier-Stokes Burgers' equation inhomogeneous Helmholtz
Metrics: PDE Error relative error absolute error SWD MMD
Baselines: EDM DiffusionPDE CoCoGen PIDM
Builds on: Diff-Instruct EDM DiffusionPDE ControlNet
Closest papers by meaning
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.92)
- DiffusionPDE: Generative PDE-Solving under Partial Observation (NeurIPS 2024 · similarity 0.90)
- Adaptive Stochastic Coefficients for Accelerating Diffusion Sampling (NeurIPS 2025 · similarity 0.88)
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.88)
- Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space (ICML 2025 · similarity 0.87)
- M2PDE: Compositional Generative Multiphysics and Multi-component PDE Simulation (ICML 2025 · similarity 0.87)
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