Graph Neural Regularizers for PDE Inverse Problems

William Lauga, James Rowbottom, Alexander Denker, Zeljko Kereta, Moshe Eliasof, Carola-Bibiane Schönlieb

Keywords: Graph Neural Networks, Inverse Problems, Neural Operator, Learned Regularizer, Partial Differential Equations

TL;DR: A physics-informed framework that exploits graph neural networks to learn a regularizer for PDE-based inverse problems

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 black-box approaches. Numerical experiments demonstrate that our framework outperforms classical regularization techniques and achieves accurate reconstructions even in highly ill-posed scenarios.

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Submission Number: 10