Graph Neural Regularizers for PDE Inverse Problems

TL;DR

This paper introduces a novel framework combining FEM-based PDE inversion with graph neural network regularization to effectively solve ill-posed inverse problems across various geometries and PDEs.

Contribution

It proposes using physics-inspired graph neural networks as learned regularizers within an iterative FEM-based inversion scheme, enhancing robustness and interpretability.

Findings

Outperforms classical regularization methods in numerical experiments.

Achieves accurate reconstructions in highly ill-posed scenarios.

Demonstrates applicability to diverse geometries and PDEs.

Abstract

We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs), where the target coefficients of the forward operator are recovered through an iterative regularization scheme that alternates between FEM-based inversion and learned graph neural regularization. The forward problem is numerically solved using the finite element method (FEM), enabling applicability to a wide range of geometries and PDEs. By leveraging the graph structure inherent to FEM discretizations, we employ physics-inspired graph neural networks as learned regularizers, providing a robust, interpretable, and generalizable alternative to standard approaches. Numerical experiments demonstrate that our framework outperforms classical regularization techniques and achieves accurate reconstructions even in highly ill-posed scenarios.