Graph Neural Regularizers for PDE Inverse Problems

2025-10-23

AI Summary

This work addresses ill-posed inverse problems governed by partial differential equations (PDEs). We propose a physics-informed, data-driven iterative regularization method that integrates physical modeling with deep learning. Specifically, we construct a graph structure via finite-element discretization, embed the forward operator into a graph neural network (GNN) framework, and design a physically interpretable GNN-based regularizer. During iteration, coefficient reconstruction and the regularization prior are jointly optimized. Unlike conventional Tikhonov or total variation regularization, our approach achieves robustness, generality, and interpretability without requiring strong prior assumptions. Experiments demonstrate significantly improved reconstruction accuracy over classical methods under highly ill-conditioned settings and low signal-to-noise ratios. The method establishes a novel paradigm for synergistic data–physics integration in solving PDE-constrained inverse problems.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Graph-based Machine Learning
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Large pretrained models with web data

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.

Problem

• Solving ill-posed inverse problems governed by PDEs
• Recovering target coefficients using iterative regularization
• Employing graph neural networks as learned regularizers

Innovation

• FEM-based inversion for PDE coefficient recovery
• Graph neural networks as learned regularizers
• Physics-inspired regularization for ill-posed problems