A deep First-Order System Least Squares method for solving elliptic PDEs

We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The we is capable of dealing with either variational and non-variational problems, because its meshless nature, it can also deal problems posed in high-dimensional domains. prove the Γ-convergence neural network approximation towards solution continuous problem extend convergence proof to some well-known related methods. Finally, present several numerical examples illustrating performance our discretization.