Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme

Xi’an Li, Jinran Wu, Lei Zhang () and Xin Tai
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Xi’an Li: Ceyear Technology Co., Ltd., Qingdao 266000, China
Jinran Wu: School of Mathematical Sciences, Queensland University of Technology, Brisbane 4001, Australia
Lei Zhang: School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
Xin Tai: Ceyear Technology Co., Ltd., Qingdao 266000, China

Mathematics, 2022, vol. 10, issue 22, 1-16

Abstract: Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson’s problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases.

Keywords: biharmonic equation; coupled scheme; DNN; variational form; Fourier mapping (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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