Graph Neural Regularizers for PDE Inverse Problems
This addresses inverse problems in computational science and engineering, offering a robust and interpretable alternative to standard methods, though it appears incremental as it builds on existing FEM and neural network approaches.
The paper tackles ill-posed inverse problems governed by PDEs by developing a framework that combines FEM-based inversion with graph neural regularization, achieving accurate reconstructions that outperform classical techniques in highly ill-posed scenarios.
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.