# Graph Neural Regularizers for PDE Inverse Problems

arXiv: https://arxiv.org/abs/2510.21012 · HTML: https://mlsanity.com/paper/2510.21012/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 35 references that are in the map.
- First posted on arXiv 23 Oct 2025.

## Builds on (most cited first, 20 of 20)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks](https://mlsanity.com/paper/2206.05437/) (2022)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Neural-network-based regularization methods for inverse problems in imaging](https://mlsanity.com/paper/2312.14849/) (2023)
- [Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems](https://mlsanity.com/paper/2103.15138/) (2021)
- [DRIP: Deep Regularizers for Inverse Problems](https://mlsanity.com/paper/2304.00015/) (2023)
- [Graph Neural Reaction Diffusion Models](https://mlsanity.com/paper/2406.10871/) (2024)
- [Recurrent Generative Adversarial Networks for Proximal Learning and Automated Compressive Image Recovery](https://mlsanity.com/paper/1711.10046/) (2017)
- [Domain independent post-processing with graph U-nets: Applications to Electrical Impedance Tomographic Imaging](https://mlsanity.com/paper/2305.05020/) (2023)
- [Learning Regularization for Graph Inverse Problems](https://mlsanity.com/paper/2408.10436/) (2024)
- [Deep Learning Based Reconstruction Methods for Electrical Impedance Tomography](https://mlsanity.com/paper/2508.06281/) (2025)

## Built on by (most cited first, 1 of 1)

- [Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography](https://mlsanity.com/paper/2605.19621/) (2026)

## Abstract (arXiv)

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.