# Recurrent Localization Networks applied to the Lippmann-Schwinger Equation

arXiv: https://arxiv.org/abs/2102.00063 · HTML: https://mlsanity.com/paper/2102.00063/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 65 references that are in the map.
- First posted on arXiv 29 Jan 2021.

## Builds on (most cited first, 19 of 19)

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- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size](https://mlsanity.com/paper/1602.07360/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [Analysis of Explainers of Black Box Deep Neural Networks for Computer Vision: A Survey](https://mlsanity.com/paper/1911.12116/) (2019)
- [A Comprehensive guide to Bayesian Convolutional Neural Network with Variational Inference](https://mlsanity.com/paper/1901.02731/) (2019)
- [Recurrent Inference Machines for Solving Inverse Problems](https://mlsanity.com/paper/1706.04008/) (2017)
- [A numerical method for computing the overall response of nonlinear composites with complex microstructure](https://mlsanity.com/paper/2012.08962/) (2020)
- [Metaheuristic Design of Feedforward Neural Networks: A Review of Two Decades of Research](https://mlsanity.com/paper/1705.05584/) (2017)

## Built on by (most cited first, 3 of 3)

- [An Introduction to Kernel and Operator Learning Methods for Homogenization by Self-consistent Clustering Analysis](https://mlsanity.com/paper/2212.00802/) (2022)
- [Refining Amortized Posterior Approximations using Gradient-Based Summary Statistics](https://mlsanity.com/paper/2305.08733/) (2023)
- [Equilibrium Conserving Neural Operators for Super-Resolution Learning](https://mlsanity.com/paper/2504.13422/) (2025)

## Abstract (arXiv)

The bulk of computational approaches for modeling physical systems in materials science derive from either analytical (i.e. physics based) or data-driven (i.e. machine-learning based) origins. In order to combine the strengths of these two approaches, we advance a novel machine learning approach for solving equations of the generalized Lippmann-Schwinger (L-S) type. In this paradigm, a given problem is converted into an equivalent L-S equation and solved as an optimization problem, where the optimization procedure is calibrated to the problem at hand. As part of a learning-based loop unrolling, we use a recurrent convolutional neural network to iteratively solve the governing equations for a field of interest. This architecture leverages the generalizability and computational efficiency of machine learning approaches, but also permits a physics-based interpretation. We demonstrate our learning approach on the two-phase elastic localization problem, where it achieves excellent accuracy on the predictions of the local (i.e., voxel-level) elastic strains. Since numerous governing equations can be converted into an equivalent L-S form, the proposed architecture has potential applications across a range of multiscale materials phenomena.