ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equation

TL;DR

This paper introduces an unsupervised, ODE-based diffusion posterior sampling method for solving inverse PDE problems, improving efficiency and robustness without requiring paired data or retraining.

Contribution

The paper presents a novel unsupervised inversion approach using ODE-based diffusion posterior sampling within a Bayesian framework for PDE inverse problems.

Findings

Efficient and robust inversion across various PDEs.

Does not require paired data or retraining.

Operates within a Bayesian inversion framework.

Abstract

In recent years we have witnessed a growth in mathematics for deep learning, which has been used to solve inverse problems of partial differential equations (PDEs). However, most deep learning-based inversion methods either require paired data or necessitate retraining neural networks for modifications in the conditions of the inverse problem, significantly reducing the efficiency of inversion and limiting its applicability. To overcome this challenge, in this paper, leveraging the score-based generative diffusion model, we introduce a novel unsupervised inversion methodology tailored for solving inverse problems arising from PDEs. Our approach operates within the Bayesian inversion framework, treating the task of solving the posterior distribution as a conditional generation process achieved through solving a reverse-time stochastic differential equation. Furthermore, to enhance the accuracy of inversion results, we propose an ODE-based Diffusion Posterior Sampling inversion algorithm. The algorithm stems from the marginal probability density functions of two distinct forward generation processes that satisfy the same Fokker-Planck equation. Through a series of experiments involving various PDEs, we showcase the efficiency and robustness of our proposed method.