Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography

TL;DR

This paper introduces a novel graph-based diffusion posterior sampling framework for solving nonlinear inverse problems like electrical impedance tomography, improving stability and accuracy over existing methods.

Contribution

It extends diffusion posterior sampling to unstructured mesh data and incorporates explicit regularization, advancing inverse problem solutions for PDEs.

Findings

RDPS produces stable, physically plausible reconstructions.

The method generalizes well to new geometries and noise conditions.

Outperforms state-of-the-art solvers in accuracy and artifact reduction.

Abstract

Deep generative models have emerged as state-of-the-art for solving inverse problems, but applying them to inverse problems for PDEs, like electrical impedance tomography (EIT) remains challenging. Because physical domains are naturally discretized as unstructured meshes rather than regular grids, standard convolutional architectures are often inadequate. In this paper, we propose a novel framework that extends diffusion posterior sampling (DPS) to graph-structured data. We develop an unconditional score-based diffusion model directly on a 2D triangular mesh to learn an accurate prior over the physical solution space. Furthermore, we introduce a regularized variant, RDPS, which incorporates explicit regularization terms, such as total variation and generalized Tikhonov, to complement the implicit diffusion prior and mitigate severe ill-posedness. Extensive experiments on synthetic and real 2D EIT datasets demonstrate that RDPS produces stable, physically plausible reconstructions. Our approach generalizes well to out-of-distribution inclusion geometries, is highly robust to measurement noise, and outperforms current state-of-the-art solvers (e.g., GPnP-BM3D, DP-SGS) in reconstruction accuracy and artifact reduction.