Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: General Discontinuous Interface

We studied the least-squares ReLU neural network method (LSNN) for solving linear advection-reaction equation with discontinuous solution in [Cai, Zhiqiang, Jingshuang Chen, and Min Liu. "Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation." Journal of Computational Physics 443 (2021): 110514]. The method is based on the least-squares formulation and employs a new class of approximating functions: multilayer perceptrons with the rectified linear unit (ReLU) activation function, i.e., ReLU deep neural networks (DNNs). In this paper, we first show that ReLU DNN with depth $\lceil \log_2(d+1)\rceil+1$ can approximate any $d$-dimensional step function on arbitrary discontinuous interfaces with any prescribed accuracy. By decomposing the solution into continuous and discontinuous parts, we prove theoretically that discretization error of the LSNN method using DNN with depth $\lceil \log_2(d+1)\rceil+1$ is mainly determined by the continuous part of the solution provided that the solution jump is constant. Numerical results for both two and three dimensional problems with various discontinuous interfaces show that the LSNN method with enough layers is accurate and does not exhibit the common Gibbs phenomena along interfaces.