Physics--informed neural networks (PINN) have shown their potential in solving both direct and inverse problems of partial differential equations. In this paper, we introduce a PINN-based deep learning approach to reconstruct one-dimensional rough surfaces from field data illuminated by an electromagnetic incident wave. In the proposed algorithm, the rough surface is approximated by a neural network, with which the spatial derivatives of surface function can be obtained via automatic differentiation and then the scattered field can be calculated via the method of moments. The neural network is trained by minimizing the loss between the calculated and the observed field data. Furthermore, the proposed method is an unsupervised approach, independent of any surface data, rather only the field data is used. Both TE field (Dirichlet boundary condition) and TM field (Neumann boundary condition) are considered. Two types of field data are used here: full scattered field data and phaseless total field data. The performance of the method is verified by testing with Gaussian-correlated random rough surfaces. Numerical results demonstrate that the PINN-based method can recover rough surfaces with great accuracy and is robust with respect to a wide range of problem regimes.

翻译：物理信息神经网络（PINN）已在求解偏微分方程的正向和逆向问题中展现出潜力。本文提出一种基于PINN的深度学习方法，用于从电磁入射波照射下的场数据重建一维粗糙表面。在该算法中，粗糙表面通过神经网络近似，利用自动微分可获取表面函数的空间导数，进而通过矩量法计算散射场。通过最小化计算场数据与观测场数据之间的损失函数来训练神经网络。此外，该方法为无监督方法，完全独立于任何表面数据，仅使用场数据。研究考虑了TE场（狄利克雷边界条件）和TM场（诺伊曼边界条件）两种情形，并采用两类场数据：全散射场数据和无相位总场数据。通过高斯相关随机粗糙表面对方法的性能进行验证。数值结果表明，该PINN方法能以高精度恢复粗糙表面，并在广泛的问题参数区间内保持鲁棒性。