Physics-informed diffusion models in spectral space
This work addresses forward and inverse PDE problems for applications like fluid dynamics, offering a more efficient method for sparse observations, though it builds incrementally on existing diffusion and physics-informed approaches.
The authors tackled the problem of generating solutions for parametric partial differential equations (PDEs) with partial observations by combining generative latent diffusion models with physics-informed machine learning in spectral space, achieving improved accuracy and computational efficiency compared to state-of-the-art diffusion-based PDE solvers.
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.