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Graph Neural Regularizers for PDE Inverse Problems

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

TL;DR

The authors address ill-posed PDE inverse problems by integrating a forward solver based on the finite element method with a learned graph neural regularizer defined on the FEM mesh, yielding a variational framework that preserves physical interpretability. They formulate a lifted, multi-step objective that includes a kinetic term and a potential term Φ implemented via GNNs, and optimize it through unrolled iterations alternating with a data-fidelity CG step. Two physics-inspired GNN architectures, GRAND and ACMP, are evaluated within this GRIP framework, demonstrating favorable performance and strong data-consistency while using orders of magnitude fewer parameters than common deep learning baselines. Across Poisson, inverse wave scattering, and electrical impedance tomography, GRIP often outperforms classical regularization and remains competitive with state-of-the-art methods, highlighting the practical value of mesh-native, physics-aligned regularizers for irregular domains. The approach offers interpretability through its variational structure and computational efficiency by operating directly on FEM meshes without interpolation.

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.

Graph Neural Regularizers for PDE Inverse Problems

TL;DR

The authors address ill-posed PDE inverse problems by integrating a forward solver based on the finite element method with a learned graph neural regularizer defined on the FEM mesh, yielding a variational framework that preserves physical interpretability. They formulate a lifted, multi-step objective that includes a kinetic term and a potential term Φ implemented via GNNs, and optimize it through unrolled iterations alternating with a data-fidelity CG step. Two physics-inspired GNN architectures, GRAND and ACMP, are evaluated within this GRIP framework, demonstrating favorable performance and strong data-consistency while using orders of magnitude fewer parameters than common deep learning baselines. Across Poisson, inverse wave scattering, and electrical impedance tomography, GRIP often outperforms classical regularization and remains competitive with state-of-the-art methods, highlighting the practical value of mesh-native, physics-aligned regularizers for irregular domains. The approach offers interpretability through its variational structure and computational efficiency by operating directly on FEM meshes without interpolation.

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.
Paper Structure (25 sections, 19 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 25 sections, 19 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Solving PDE Inverse Problems with Graph Neural Regularizers
  3. GNN Architectures
  4. Experiments and Results
  5. Conclusion
  6. Related Work
  7. PDE Descriptions and Details
  8. Linearization.
  9. Poisson Equation
  10. Data generation.
  11. FEM and weak formulation.
  12. Inverse Wave Scattering (Helmholtz Equation)
  13. Boundary conditions
  14. Coefficient distribution.
  15. FEM and weak formulation.
  16. ...and 10 more sections

Figures (3)

  • Figure 1: Poisson (sparse) reconstruction examples
  • Figure 2: Inverse Wave Scattering reconstruction examples
  • Figure 3: EIT reconstruction examples