Graph Neural Regularizers for PDE Inverse Problems

• URL: http://arxiv.org/abs/2510.21012v1
• Date: Thu, 23 Oct 2025 21:43:25 GMT
• Title: Graph Neural Regularizers for PDE Inverse Problems
• Authors: William Lauga, James Rowbottom, Alexander Denker, Željko Kereta, Moshe Eliasof, Carola-Bibiane Schönlieb,
• Abstract summary: We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs)<n>The forward problem is numerically solved using the finite element method (FEM)<n>We employ physics-inspired graph neural networks as learned regularizers, providing a robust, interpretable, and generalizable alternative to standard approaches.
• Score: 62.49743146797144
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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