Particle-Guided Diffusion Models for Partial Differential Equations

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.

Solving partial differential equations (PDEs) with diffusion models is a new and challenging question. However, since diffusion models are typically data-focused and are trained from simulations of PDEs, directly sampling from the diffusion model ignores the PDE contraints. In this paper, we develop a sequential Monte Carlo sampler to incorporate the PDE contraints at inference time with such diffusion models, in a statisticall principled ways. Empirical results show that the performance largely excel the baselines in a variety of bechmarking problems for both forward and inverse PDE problems.