Physics-informed diffusion models in spectral space

TL;DR

This paper introduces a physics-informed spectral diffusion model that efficiently generates PDE solutions conditioned on partial data, improving accuracy and speed over existing methods for complex equations.

Contribution

It combines spectral space diffusion with physics constraints, enabling reduced-dimensionality PDE solution generation with enhanced accuracy and efficiency.

Findings

Outperforms existing diffusion-based PDE solvers on benchmark problems

Achieves significant dimensionality reduction in spectral space

Demonstrates improved computational efficiency and accuracy

Abstract

We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier--Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at https://github.com/deeplearningmethods/PISD.