  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.
