We studied the least-squares ReLU neural network (LSNN) method 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 properly designed and physics preserved discrete differential operator. In this paper, we study the LSNN method for problems with discontinuity 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 a discontinuity interface generated by a vector field as streamlines 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 test problems with various discontinuity interfaces show that the LSNN method with enough layers is accurate and does not exhibit the common Gibbs phenomena along discontinuity interfaces.

翻译：本文研究了利用最小二乘ReLU神经网络（LSNN）方法求解具有间断解的线性对流-反应方程，相关工作可参见[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]。该方法基于最小二乘形式，并采用一类新型逼近函数：ReLU神经网络（NN）函数。与其他基于神经网络的方法不同，LSNN方法的关键附加组成部分是引入了一个经过恰当设计且保持物理特性的离散微分算子。本文针对具有间断界面的问题研究了LSNN方法。首先，我们证明深度为$\lceil \log_2(d+1)\rceil+1$的ReLU神经网络函数能够以任意指定精度逼近由向量场形成的流线所生成的$d$维间断界面上的任意阶梯函数。通过将解分解为连续部分和间断部分，我们从理论上证明：在解的跳跃值为常数的条件下，使用深度为$\lceil \log_2(d+1)\rceil+1$的ReLU神经网络函数的LSNN方法的离散误差主要由解的连续部分决定。针对二维和三维测试问题中各类间断界面的数值结果表明，具有足够层数的LSNN方法精度良好，且沿间断界面未出现常见的吉布斯现象。