{"ID":2851398,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.21012","arxiv_id":"2510.21012","title":"Graph Neural Regularizers for PDE 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 approaches. Numerical experiments demonstrate that our framework outperforms classical regularization techniques and achieves accurate reconstructions even in highly ill-posed scenarios.","short_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 regulariza...","url_abs":"https://arxiv.org/abs/2510.21012","url_pdf":"https://arxiv.org/pdf/2510.21012v1","authors":"[\"William Lauga\",\"James Rowbottom\",\"Alexander Denker\",\"Željko Kereta\",\"Moshe Eliasof\",\"Carola-Bibiane Schönlieb\"]","published":"2025-10-23T21:43:25Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.LG\"]","methods":"[\"Graph Neural Network\"]","has_code":false}