High Order Deep Neural Network for Solving High Frequency Partial Differential Equations (Communications in Computational Physics, 2022)
This paper proposes a high order deep neural network (HOrderDNN) forsolving high frequency partial differential equations (PDEs), which incorporates theidea of "high order" from finite element methods (FEMs) into commonly-used deepneural networks (DNNs) to obtain greater approximation ability. The main idea ofHOrderDNN is introducing a nonlinear transformation layer between the input layerand the first hidden layer to form a high order polynomial space with the degree notexceeding $p$, followed by a normal DNN. The order $p$ can be guided by the regularityof solutions of PDEs. The performance of HOrderDNN is evaluated on high frequencyfunction fitting problems and high frequency Poisson and Helmholtz equations. Theresults demonstrate that: HOrderDNNs($p > 1$) can efficiently capture the high frequency information in target functions; and when compared to physics-informed neural network (PINN), HOrderDNNs($p > 1$) converge faster and achieve much smallerrelative errors with same number of trainable parameters. In particular, when solvingthe high frequency Helmholtz equation in Section 3.5, the relative error of PINN staysaround 1 with its depth and width increase, while the relative error can be reduced toaround 0.02 as $p$ increases (see Table 5).