# Solving Maxwell's Equation in 2D with Neural Networks with Local Converging Inputs

arXiv: https://arxiv.org/abs/2302.02860 · HTML: https://mlsanity.com/paper/2302.02860/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 6 Feb 2023.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data](https://mlsanity.com/paper/2109.12482/) (2021)
- [Neural-net-induced Gaussian process regression for function approximation and PDE solution](https://mlsanity.com/paper/1806.11187/) (2018)
- [Machine learning for protein folding and dynamics](https://mlsanity.com/paper/1911.09811/) (2019)
- [Data-driven Identification of 2D Partial Differential Equations using extracted physical features](https://mlsanity.com/paper/2010.10626/) (2020)
- [Neural network augmented wave-equation simulation](https://mlsanity.com/paper/1910.00925/) (2019)

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

- [Neural Network with Local Converging Input (NNLCI) for Supersonic Flow Problems with Unstructured Grids](https://mlsanity.com/paper/2310.15299/) (2023)
- [A PNP ion channel deep learning solver with local neural network and finite element input data](https://mlsanity.com/paper/2401.17513/) (2024)
- [Neural Networks with Local Converging Inputs for Efficient Options Pricing Models](https://mlsanity.com/paper/2608.02778/) (2026)

## Abstract (arXiv)

In this paper we apply neural networks with local converging inputs (NNLCI), originally introduced in [arXiv:2109.09316], to solve the two dimensional Maxwell's equation around perfect electric conductors (PECs). The input to the networks consist of local patches of low cost numerical solutions to the equation computed on two coarse grids, and the output is a more accurate solution at the center of the local patch. We apply the recently developed second order finite difference method [arXiv:2209.00740] to generate the input and training data which captures the scattering of electromagnetic waves off of a PEC at a given terminal time. The advantage of NNLCI is that once trained it offers an efficient alternative to costly high-resolution conventional numerical methods; our numerical experiments indicate the computational complexity saving by a factor of 83 in terms of the number of spatial-temporal grid points. In contrast with existing research work on applying neural networks to directly solve PDEs, our method takes advantage of the local domain of dependence of the Maxwell's equation in the input solution patches, and is therefore simpler, yet still robust. We demonstrate that we can train our neural network on some PECs to predict accurate solutions to different PECs with quite different geometries from any of the training examples.