Title: Physics-informed diffusion models in spectral space

Abstract: This paper introduces Physics-Informed Spectral Diffusion (PISD), a novel framework that integrates generative latent diffusion models with physics-informed machine learning. This approach is designed to generate solutions for partial differential equations (PDEs) given partial observations, effectively addressing both forward and inverse PDE challenges. By utilizing a latent space composed of scaled spectral representations, we model the joint distribution of PDE parameters and their corresponding solutions through a diffusion process. In this context, Gaussian noise is associated with functions exhibiting controlled regularity. This spectral strategy offers substantial dimensionality reduction relative to traditional grid-based diffusion models and guarantees that the resulting process in function space stays within the domain where PDE operators are well-defined. Leveraging diffusion posterior sampling, we incorporate physics-based constraints and measurement conditions during the inference phase, utilizing Adam-based updates at every diffusion step. Our evaluation of the method on the Poisson, Helmholtz, and incompressible Navier-Stokes equations reveals enhanced accuracy and computational efficiency when compared to current state-of-the-art diffusion-based PDE solvers, particularly in scenarios involving sparse observations. The source code can be accessed at https://github.com/deeplearningmethods/PISD.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC