A Novel Deep Neural Network Algorithm for the Helmholtz Scattering Problem In the Unbounded Domain

Abstract

. In this paper, we develop a novel meshless, ray-based deep neural network algorithm for solving the high-frequency Helmholtz scattering problem in the unbounded domain. While our recent work [44] designed a deep neural network method for solving the Helmholtz equation over (cid:12)nite bounded domains, this paper deals with the more general and di(cid:14)cult case of unbounded regions. By using the perfectly matched layer method, the original mathematical model in the unbounded domain is transformed into a new format of second-order system in a (cid:12)nite bounded domain with simple homogeneous Dirichlet boundary conditions. Compared with the Helmholtz equation in the bounded domain, the new system is equipped with variable coe(cid:14)cients. Then, a deep neural network algorithm is designed for the new system, where the rays in various random directions are used as the basis of the numerical solution. Various numerical examples have been carried out to demonstrate the accuracy and e(cid:14)ciency of the proposed numerical method. The proposed method has the advantage of easy implementation and meshless while maintaining high accuracy. To the best of the author’s knowledge, this is the (cid:12)rst deep neural network method to solve the Helmholtz equation in the unbounded domain.