Universal Approximation Theorem and Deep Learning for the Solution of Frequency-Domain Electromagnetic Scattering Problems

Unlike the universal approximation theorems for functions mapping from a real-valued (RV) vector to an RV number or from a complex-valued (CV) vector to a CV number, in the field of electromagnetism, we need to approximate functions mapping from an RV vector to a CV number when we consider the electric field as a function of the spatial coordinate in the frequency domain. Typically, CV numbers contain phase information. When such phase information is handled properly, the performance of the neural networks (NNs) can be improved. This work derives a universal approximation theorem for functions mapping from an RV vector to a CV number. A deep NN, named HV-DL, is designed accordingly, which consists of an RV input layer, an RV module containing two branches, a CV module, and a CV output layer. The proposed universal approximation theorem is verified by numerical experiments on the HV-DL solution of the 2-D electric field integral equation (EFIE). To integrate the underlying physics of electromagnetic (EM) scattering into the proposed HV-DL, the loss function is computed according to the EFIE.