The least-squares ReLU neural network method (LSNN) was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai2023least}. The method is based on an equivalent least-squares formulation and employs ReLU neural network (NN) functions with a $\lceil \log_2(d+1)\rceil+1$ layer representation for approximating the solution. In this paper, we show theoretically that the method is also capable of approximating a non-constant jump along the discontinuous interface of the underlying problem that is not necessarily a straight line. Numerical results for test problems with various non-constant jumps and interfaces show that the LSNN method with $\lceil \log_2(d+1)\rceil+1$ layers approximates the solution accurately with DoFs less than that of mesh-based methods and without the common Gibbs phenomena along the discontinuous interface.

翻译：暂无翻译