Physics-informed diffusion models in spectral space

2026-02-10

AI Summary

This work proposes a generative diffusion model based on a spectral representation latent space to address the generation of solutions to parametric partial differential equations (PDEs) under partial observations, unifying both forward and inverse PDE tasks. The method models the joint distribution of PDE parameters and solutions in a compressed, regularized spectral latent space, incorporating physical constraints and posterior sampling via diffusion to assimilate observational data during inference while ensuring solutions reside within the well-posed function class defined by the PDE operator. Each sampling step integrates Adam optimization to enhance numerical stability. Experiments on Poisson, Helmholtz, and incompressible Navier–Stokes equations demonstrate that the approach significantly outperforms existing diffusion-based PDE solvers under sparse observation conditions, achieving notable advances in both accuracy and computational efficiency.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• partial differential equations
• inverse problems
• generative modeling
• spectral methods
• physics-informed learning

Innovation

• physics-informed diffusion
• spectral representation
• latent diffusion models
• parametric PDEs
• posterior sampling