  This paper studies least-squares ReLU neural network method for solving the
linear advection-reaction problem with discontinuous solution. The method is a
discretization of an equivalent least-squares formulation in the set of neural
network functions with the ReLU activation function. The method is capable of
approximating the discontinuous interface of the underlying problem
automatically through the free hyper-planes of the ReLU neural network and,
hence, outperforms mesh-based numerical methods in terms of the number of
degrees of freedom. Numerical results of some benchmark test problems show that
the method can not only approximate the solution with the least number of
parameters, but also avoid the common Gibbs phenomena along the discontinuous
interface. Moreover, a three-layer ReLU neural network is necessary and
sufficient in order to well approximate a discontinuous solution with an
interface in $\mathbb{R}^2$ that is not a straight line.
