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A least-squares neural network (LSNN) method was introduced for solvingscalar linear and nonlinear hyperbolic conservation laws (HCLs) in [7, 6]. Thismethod is based on an equivalent least-squares (LS) formulation and uses ReLUneural network as approximating functions, making it ideal for approximatingdiscontinuous functions with unknown interface location. In the design of theLSNN method for HCLs, the numerical approximation of differential operators isa critical factor, and standard numerical or automatic differentiation alongcoordinate directions can often lead to a failed NN-based method. To overcomethis challenge, this paper rewrites HCLs in their divergence form of space andtime and introduces a new discrete divergence operator. As a result, theproposed LSNN method is free of penalization of artificial viscosity.Theoretically, the accuracy of the discrete divergence operator is estimatedeven for discontinuous solutions. Numerically, the LSNN method with the newdiscrete divergence operator was tested for several benchmark problems withboth convex and non-convex fluxes, and was able to compute the correct physicalsolution for problems with rarefaction, shock or compound waves. The method iscapable of capturing the shock of the underlying problem without oscillation orsmearing, even without any penalization of the entropy condition, totalvariation, and/or artificial viscosity.

Least-Squares Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Laws: Discrete Divergence Operator