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 a least-squares formulation and uses a new class of approximating functions: ReLU neural network (NN) functions. A critical and additional component of the LSNN method, differing from other NN-based methods, is the introduction of a proper designed discrete differential operator. In this paper, we study the LSNN method for problems with arbitrary discontinuous interfaces. First, we show that ReLU NN functions 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 ReLU NN functions 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 the discontinuous interface.

翻译：我们研究了文献[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]中提出的最小二乘ReLU神经网络方法，用于求解具有间断解的线性平流反应方程。该方法基于最小二乘公式，并采用一类新的逼近函数：ReLU神经网络函数。与其他基于神经网络的方法不同，LSNN方法的一个关键且额外的组成部分是引入了一个精心设计的离散微分算子。本文研究了该方法在任意间断界面问题中的应用。首先，我们证明，对于任意间断界面上的任意d维阶跃函数，深度为$\lceil \log_2(d+1)\rceil+1$的ReLU神经网络函数可以任意精度逼近。通过将解分解为连续部分和间断部分，我们从理论上证明，在解跳跃为常数的情况下，使用深度为$\lceil \log_2(d+1)\rceil+1$的ReLU神经网络函数的LSNN方法的离散化误差主要由解的连续部分决定。针对二维和三维问题中各种间断界面的数值结果表明，具有足够层数的LSNN方法准确且沿间断界面未出现常见的吉布斯现象。