TY - JOUR
T1 - A Novel Deep Neural Network Algorithm for the Helmholtz Scattering Problem in the Unbounded Domain
AU - Yang , Andy L
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 724
EP - 738
PY - 2023
DA - 2023/09
SN - 20
DO - http://doi.org/ 10.4208/ijnam2023-1032
UR - https://global-sci.org/intro/article_detail/ijnam/22010.html
KW - Deep Learning, plane wave, deep neural network, loss, high frequency, Helmholtz equation.
AB - <p style="text-align: justify;">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
finite bounded domains, this paper deals with the more general and difficult 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 finite bounded
domain with simple homogeneous Dirichlet boundary conditions. Compared with the Helmholtz
equation in the bounded domain, the new system is equipped with variable coefficients. 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 efficiency 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 first deep neural network method to
solve the Helmholtz equation in the unbounded domain.</p>