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.
@article{arxiv.2601.23262,
title = {Particle-Guided Diffusion Models for Partial Differential Equations},
author = {Andrew Millard and Fredrik Lindsten and Zheng Zhao},
journal= {arXiv preprint arXiv:2601.23262},
year = {2026}
}