Particle-Guided Diffusion Models for Partial Differential Equations
This work addresses the challenge of ensuring physical consistency in generative models for PDEs, which is crucial for applications in scientific computing and engineering, though it appears incremental as it builds on existing diffusion models with novel guidance.
The authors tackled the problem of generating physically admissible solutions to partial differential equations (PDEs) by introducing a guided stochastic sampling method that incorporates physics-based guidance from PDE residuals and observational constraints, resulting in lower numerical error across multiple benchmark and complex PDE systems compared to existing state-of-the-art generative methods.
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.