# Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems

arXiv: https://arxiv.org/abs/2103.15138 · HTML: https://mlsanity.com/paper/2103.15138/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 28 Mar 2021.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [A deep convolutional neural network using directional wavelets for low-dose X-ray CT reconstruction](https://mlsanity.com/paper/1610.09736/) (2016)
- [Computed Tomography Reconstruction Using Deep Image Prior and Learned Reconstruction Methods](https://mlsanity.com/paper/2003.04989/) (2020)
- [Model based learning for accelerated, limited-view 3D photoacoustic tomography](https://mlsanity.com/paper/1708.09832/) (2017)
- [Recurrent Inference Machines for Solving Inverse Problems](https://mlsanity.com/paper/1706.04008/) (2017)
- [Deep D-bar: Real time Electrical Impedance Tomography Imaging with Deep Neural Networks](https://mlsanity.com/paper/1711.03180/) (2017)
- [GResNet: Graph Residual Network for Reviving Deep GNNs from Suspended Animation](https://mlsanity.com/paper/1909.05729/) (2019)
- [Multi-Scale Learned Iterative Reconstruction](https://mlsanity.com/paper/1908.00936/) (2019)
- [Beltrami-Net: Domain Independent Deep D-bar Learning for Absolute Imaging with Electrical Impedance Tomography (a-EIT)](https://mlsanity.com/paper/1811.12830/) (2018)
- [An efficient Quasi-Newton method for nonlinear inverse problems via learned singular values](https://mlsanity.com/paper/2012.07676/) (2020)

## Built on by (most cited first, 10 of 10)

- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Physics Embedded Machine Learning for Electromagnetic Data Imaging](https://mlsanity.com/paper/2207.12607/) (2022)
- [A Comparative Study of Variational Autoencoders, Normalizing Flows, and Score-based Diffusion Models for Electrical Impedance Tomography](https://mlsanity.com/paper/2310.15831/) (2023)
- [Domain independent post-processing with graph U-nets: Applications to Electrical Impedance Tomographic Imaging](https://mlsanity.com/paper/2305.05020/) (2023)
- [Enhancing Electrical Impedance Tomography reconstruction using Learned Half-Quadratic Splitting Networks with Anderson Acceleration](https://mlsanity.com/paper/2304.14491/) (2023)
- [Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approaches](https://mlsanity.com/paper/2310.18636/) (2023)
- [Inverse Problem of Nonlinear Schr"odinger Equation as Learning of Convolutional Neural Network](https://mlsanity.com/paper/2107.08593/) (2021)
- [Physics-Driven Neural Compensation For Electrical Impedance Tomography](https://mlsanity.com/paper/2504.18067/) (2025)
- [Graph Neural Regularizers for PDE Inverse Problems](https://mlsanity.com/paper/2510.21012/) (2025)
- [Graph convolutional networks enable fast hemorrhagic stroke monitoring with electrical impedance tomography](https://mlsanity.com/paper/2412.07888/) (2024)

## Abstract (arXiv)

The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method typical for nonlinear inverse problems, interpolation and embeddings are needed. To overcome this, we present a flexible framework to extend model-based learning directly to nonuniform meshes, by interpreting the mesh as a graph and formulating our network architectures using graph convolutional neural networks. This gives rise to the proposed iterative Graph Convolutional Newton-type Method (GCNM), which includes the forward model in the solution of the inverse problem, while all updates are directly computed by the network on the problem specific mesh. We present results for Electrical Impedance Tomography, a severely ill-posed nonlinear inverse problem that is frequently solved via optimization-based methods, where the forward problem is solved by finite element methods. Results for absolute EIT imaging are compared to standard iterative methods as well as a graph residual network. We show that the GCNM has strong generalizability to different domain shapes and meshes, out of distribution data as well as experimental data, from purely simulated training data and without transfer training.