A Priori Error Estimate of Deep Mixed Residual Method for Elliptic PDEs

In this work, we derive a priori error estimate of the deep mixed residual method (DMRM) when solving some elliptic partial differential equations (PDEs). DMRM is a new deep-learning based method for solving PDEs and it has been shown to be efficient and accurate in previous studies. Our work is the first theoretical study of DMRM. We prove that the neural network solutions will converge if we increase the training samples and network size without any constraint on the ratio of training samples to the network size. Besides, our results suggest that the DMRM can approximate the Laplacian of the solution by the intermediate auxiliary variable, which is dismissing in the deep Ritz method. It is verified by the numerical experiments.