MIM: A deep mixed residual method for solving high-order partial differential equations

In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. Notable examples include Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries equation. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of the first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. For aforementioned classical numerical methods, the choice of trial and test functions is important for stability and accuracy issues in many cases. MIM shares this property when DNNs are employed to approximate unknowns functions in the first-order system. In one case, we use nearly the same DNN to approximate all unknown functions and in the other case, we use totally different DNNs for different unknown functions. Numerous results of MIM with different loss functions and different choices of DNNs are given for four types of PDEs. In most cases, MIM provides better approximations (not only for high-order derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. MIM with multiple DNNs often provides better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Numerical results also indicate interesting connections between MIM and classical numerical methods. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis.