DiffusionPDE: Generative PDE-Solving Under Partial Observation

TL;DR

DiffusionPDE introduces a generative diffusion model framework that effectively solves PDEs with incomplete data by jointly modeling solutions and coefficients, outperforming existing methods in forward and inverse problems.

Contribution

The paper presents a novel generative diffusion approach for PDE solving under partial observations, enabling joint modeling of solutions and coefficients.

Findings

Outperforms state-of-the-art PDE solvers on partial data

Effectively fills missing information in PDE problems

Works for both forward and inverse PDE tasks

Abstract

We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical solvers. Most existing forward or inverse PDE approaches perform poorly when the observations on the data or the underlying coefficients are incomplete, which is a common assumption for real-world measurements. In this work, we propose DiffusionPDE that can simultaneously fill in the missing information and solve a PDE by modeling the joint distribution of the solution and coefficient spaces. We show that the learned generative priors lead to a versatile framework for accurately solving a wide range of PDEs under partial observation, significantly outperforming the state-of-the-art methods for both forward and inverse directions.