LSNN Method For Scalar Nonlinear HCLs: Discrete Divergence Operator

The least-squares neural network (LSNN) method was introduced for solving scalar linear and nonlinear hyperbolic conservation laws in [6, 5]. The method is based on an equivalent least-squares (LS) formulation and employs ReLU neural network as approximating functions, that is especially suitable for approximating discontinuous functions with unknown interface location. In design of the LSNN method for HCLs, numerical approximation of differential operator plays a critical role, and standard numerical or automatic differentiation along coordinate directions usually results in a failing NN-based method. To overcome this difficulty, this paper rewrites HCLs in their divergence form of space and time and introduces a new discrete divergence operator. Theoretically, accuracy of the discrete divergence operator is estimated even if the solution is discontinuous. Numerically, the resulting LSNN method with the new discrete divergence operator is tested for several benchmark problems with both convex and non-convex fluxes; the method is capable of computing the correct physical solution for problems with rarefaction waves and capturing the shock of the underlying problem without oscillation or smearing.