Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Generating Physical Dynamics under Priors
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Incorporates distributional priors (SE(n)/permutation invariance via equivariant noise matching) and physical feasibility priors (PDE, energy and momentum conservation via penalty losses with Jensen's-gap handling) into diffusion models to generate physically feasible dynamics.
Problem
Diffusion-based generative models for physical dynamics often violate physical laws because physical priors (equations, conservation laws, symmetries) are not integrated into the generative output, and constraints do not transfer directly to conditional expectations due to Jensen's gap.
Approach
Embed distributional priors (SE(n)/permutation invariance via data augmentation and noise matching) and physical feasibility priors (PDE and conservation constraints) as penalty losses on the diffusion model's output, classifying constraints as linear, multilinear, convex, or reducible/general nonlinear to handle Jensen's gap, with auxiliary models predicting nonlinear quantities.
Key results
- Advection RMSE 1.6536 vs 1.7263 without prior (x10^-2); Darcy flow RMSE 1.9678 vs 2.0648 (x10^-2).
- Three-body energy error 0.5465 with prior vs 4.3292 without; five-spring momentum error 0.3898 vs 5.3699.
- Shallow water PDE constraint RMSE 7.7618 with prior vs 8.0153 without.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add PDE feasibility prior | Diffusion model · Advection prediction | RMSE (× 10^-2): 1.7263±0.0491 -> 1.6536±0.0677 | Helped |
| Add PDE feasibility prior | Diffusion model · Darcy flow prediction | RMSE (× 10^-2): 2.0648±0.0600 -> 1.9678±0.0651 | Helped |
| Add PDE feasibility prior | Diffusion model · Advection sample feasibility | PDE constraint RMSE: 2.398±0.024 -> 2.305±0.001 | Helped |
| Add PDE feasibility prior | Diffusion model · Burger sample feasibility | PDE constraint RMSE: 6.862±0.060 -> 6.610±0.012 | Helped |
| Add PDE feasibility prior | Diffusion model · Shallow water sample feasibility | PDE constraint RMSE: 8.0153±0.0960 -> 7.7618±0.0645 | Helped |
| Add physical feasibility priors | Diffusion model · Three-body | Traj error: 2.4132±0.1208 -> 1.9880±0.3418 | Helped |
| Add physical feasibility priors | Diffusion model · Three-body | Vel error: 2.5745±0.0790 -> 0.8328±0.1042 | Helped |
| Add physical feasibility priors | Diffusion model · Three-body | Energy error: 4.3292±0.7235 -> 0.5465±0.0705 | Helped |
| Add physical feasibility priors | Diffusion model · Five-spring | Dynamic error: 5.1754±0.0286 -> 5.0731±0.0406 | Helped |
| Add physical feasibility priors | Diffusion model · Five-spring | Momentum error: 5.3699±0.0462 -> 0.3898±0.0118 | Helped |
| Add physical feasibility priors | Diffusion model · Five-spring | Energy error: 1.0618±0.0243 -> 0.7418±0.0129 | Helped |
| Use the dataset-recommended matching objective (data matching for PDEs and noise matching for particle dynamics) rather than the alternative objective (ablation) | Diffusion model without physical feasibility priors · PDE and particle dynamics datasets | Model performance | Helped |
Surprising findings
- Directly applying a nonlinear (energy) constraint to the diffusion output without handling Jensen's gap degrades performance.
- Noise matching is preferred for invariant particle systems while data matching is preferred for smooth PDE samples, and the choice strongly affects sample quality.
Limitations
- No architecture satisfies SE(n) and permutation invariance in diffusion models, so data augmentation is used instead.
- Nonlinear constraints require auxiliary models or decompositions, and direct penalty application can hurt performance.
- Evaluation limited to synthetic datasets with relatively small-scale settings.
Experimental setup
Datasets: advection Darcy flow Burgers shallow water three-body five-spring
Benchmarks: advection Darcy flow Burgers shallow water three-body five-spring
Metrics: RMSE PDE constraint RMSE MSE trajectory error velocity error energy error momentum error
Baselines: diffusion model without priors (w/o prior) prior by PINN
Builds on: denoising diffusion probabilistic models physics-informed neural networks DPM-Solver Tweedie's formula
Closest papers by meaning
- Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints (NeurIPS 2025 · similarity 0.88)
- Composing Diffusion Priors with Explicit Physical Context via Generative Gibbs Sampling (NeurIPS 2026 · similarity 0.88)
- Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observations (NeurIPS 2025 · similarity 0.87)
- Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling (NeurIPS 2024 · similarity 0.87)
- Physics-Informed Diffusion Models (ICLR 2025 · similarity 0.87)
- Generative Modeling with Phase Stochastic Bridge (ICLR 2024 · similarity 0.87)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.