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