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 $8^3$ 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.

翻译：本文应用最初在[arXiv:2109.09316]中提出的局部收敛输入神经网络（NNLCI），求解完美电导体（PEC）周围的二维麦克斯韦方程组。网络输入由在两个粗网格上计算的方程低代价数值解的局部补丁组成，输出为局部补丁中心处更精确的解。我们采用近期开发的二阶有限差分方法[arXiv:2209.00740]生成输入和训练数据，该方法捕获了电磁波在给定终端时刻从PEC散射的过程。NNLCI的优势在于，一旦训练完成，它就能为昂贵的高分辨率传统数值方法提供高效的替代方案；我们的数值实验表明，在时空网格点数量方面，计算复杂度节省了$8^3$倍。与现有的将神经网络直接应用于求解偏微分方程的研究工作相比，我们的方法利用了麦克斯韦方程组在输入解补丁中的局部依赖域，因此更简单且依然稳健。我们证明，可以在某些PEC上训练神经网络，以预测与任何训练示例几何形状差异较大的不同PEC的精确解。