Particle-Guided Diffusion Models for Partial Differential Equations

TL;DR

This paper presents a physics-guided stochastic sampling approach integrated with diffusion models and a Sequential Monte Carlo framework to efficiently generate physically admissible solutions for complex PDE systems.

Contribution

It introduces a novel PDE solver combining diffusion models with physics-based guidance and SMC, improving accuracy over existing generative methods.

Findings

Lower numerical error on benchmark PDEs

Effective for multiphysics systems

Scalable generative PDE solution method

Abstract

We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples remain physically admissible. We embed this sampling procedure within a new Sequential Monte Carlo (SMC) framework, yielding a scalable generative PDE solver. Across multiple benchmark PDE systems as well as multiphysics and interacting PDE systems, our method produces solution fields with lower numerical error than existing state-of-the-art generative methods.