We introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of degrees of freedom. This paper studies the LSNN method for scalar nonlinear hyperbolic conservation law. The method is a discretization of an equivalent least-squares (LS) formulation in the set of neural network functions with the ReLU activation function. Evaluation of the LS functional is done by using numerical integration and conservative finite volume scheme. Numerical results of some test problems show that the method is capable of approximating the discontinuous interface of the underlying problem automatically through the free breaking lines of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.

翻译：我们引入了用不连续解决方案解决线性对冲反应问题的最小方程 ReLU神经网络(LSNN)方法, 并表明该方法在自由度方面优于基于网状的数字方法。 本文研究了用于标度非线性双曲保护法的 LSNN 方法。 该方法在神经网络功能组中是一种与RELU激活功能相当的最小方程配方(LS)的分解。 对 LS 功能的评估是通过数字集成和保守的有限体积方案完成的。 一些测试问题的数字结果显示,该方法能够通过ReLU神经网络的自由断线自动近似问题不连续的界面。 此外, 该方法在不连续的界面中并不展示常见的 Gib 现象 。