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

2026-05-19

AI Summary

This work addresses the challenge of solving nonlinear inverse problems governed by partial differential equations on unstructured grids—such as in electrical impedance tomography—where conventional convolutional generative models struggle to capture the underlying physics. The study presents the first extension of diffusion posterior sampling (DPS) to graph-structured data, introducing an unconditional score-based diffusion model trained on 2D triangular meshes to learn a physically informed prior over solution spaces. Furthermore, it proposes a regularized DPS (RDPS) framework that explicitly integrates total variation and generalized Tikhonov regularization with the implicit diffusion prior. This approach substantially mitigates the severe ill-posedness of the inverse problem, yielding stable and physically plausible reconstructions on both synthetic and real 2D electrical impedance data. The method demonstrates strong robustness to out-of-distribution geometries and measurement noise, outperforming existing techniques such as GPnP-BM3D and DP-SGS in reconstruction accuracy and artifact suppression.

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.

Problem

• nonlinear inverse problems
• electrical impedance tomography
• unstructured meshes
• ill-posedness
• PDE-based inverse problems

Innovation

• diffusion posterior sampling
• graph-structured data
• electrical impedance tomography
• score-based generative model
• regularized inverse problems