A least-squares neural network (LSNN) method was introduced for solving scalar linear and nonlinear hyperbolic conservation laws (HCLs) in [7, 6]. This method is based on an equivalent least-squares (LS) formulation and uses ReLU neural network as approximating functions, making it ideal for approximating discontinuous functions with unknown interface location. In the design of the LSNN method for HCLs, the numerical approximation of differential operators is a critical factor, and standard numerical or automatic differentiation along coordinate directions can often lead to a failed NN-based method. To overcome this challenge, this paper rewrites HCLs in their divergence form of space and time and introduces a new discrete divergence operator. As a result, the proposed LSNN method is free of penalization of artificial viscosity. Theoretically, the accuracy of the discrete divergence operator is estimated even for discontinuous solutions. Numerically, the LSNN method with the new discrete divergence operator was tested for several benchmark problems with both convex and non-convex fluxes, and was able to compute the correct physical solution for problems with rarefaction, shock or compound waves. The method is capable of capturing the shock of the underlying problem without oscillation or smearing, even without any penalization of the entropy condition, total variation, and/or artificial viscosity.

翻译：文献[7, 6]提出了一种最小二乘神经网络（LSNN）方法，用于求解标量线性和非线性双曲守恒律（HCLs）。该方法基于等效的最小二乘（LS）形式，并采用ReLU神经网络作为逼近函数，特别适用于逼近间断位置未知的不连续函数。在针对HCLs设计LSNN方法时，微分算子的数值逼近是关键因素，而沿坐标方向的标准数值微分或自动微分常导致基于NN的方法失效。为克服这一挑战，本文将HCLs重写为时空散度形式，并引入一种新的离散散度算子。由此，所提出的LSNN方法无需人工粘性惩罚项。理论上，即使对于间断解，离散散度算子的精度也可估。数值实验中，采用新离散散度算子的LSNN方法在多个涉及凸通量与非凸通量的基准问题上进行了测试，能够正确计算包含稀疏波、激波或复合波等情形下的物理解。该方法可准确捕捉激波而无振荡或抹平现象，甚至无需添加熵条件、总变差或人工粘性的惩罚项。