The least-squares ReLU neural network method (LSNN) was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai2023least}. The method is based on an equivalent least-squares formulation and employs ReLU neural network (NN) functions with a $\lceil \log_2(d+1)\rceil+1$ layer representation for approximating the solution. In this paper, we show theoretically that the method is also capable of approximating a non-constant jump along the discontinuous interface of the underlying problem that is not necessarily a straight line. Numerical results for test problems with various non-constant jumps and interfaces show that the LSNN method with $\lceil \log_2(d+1)\rceil+1$ layers approximates the solution accurately with DoFs less than that of mesh-based methods and without the common Gibbs phenomena along the discontinuous interface.

翻译：最小二乘ReLU神经网络方法（LSNN）在文献《Cai2021linear, cai2023least》中被引入并用于求解具有间断解的线性对流-反应方程。该方法基于等价的最小二乘公式，并采用具有$\lceil \log_2(d+1)\rceil+1$层表示的ReLU神经网络函数来逼近解。本文从理论上证明，该方法同样能够逼近问题间断界面上的非恒定跳跃，且该界面不必为直线。针对具有各种非恒定跳跃和界面的测试问题的数值结果表明，采用$\lceil \log_2(d+1)\rceil+1$层的LSNN方法能够以少于网格方法的自由度准确逼近解，且沿间断界面无常见的吉布斯现象。