Solving a class of high-order elliptic pdes using deep neural networks based on its coupled scheme

Journal article

Li, Xi'An, Wu, Jinran, Zhang, Lei and Tai, Xin. (2022). Solving a class of high-order elliptic pdes using deep neural networks based on its coupled scheme. Mathematics. 10(22), p. Article 4186. https://doi.org/10.3390/math10224186
AuthorsLi, Xi'An, Wu, Jinran, Zhang, Lei and Tai, Xin
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.

Keywordsbiharmonic equation; coupled scheme; DNN; variational form; Fourier mapping
Year2022
JournalMathematics
Journal citation10 (22), p. Article 4186
PublisherMultidisciplinary Digital Publishing Institute (MDPI AG)
ISSN2227-7390
Digital Object Identifier (DOI)https://doi.org/10.3390/math10224186
Scopus EID2-s2.0-85143168981
Open accessPublished as ‘gold’ (paid) open access
Page range1-16
FunderNational Natural Science Foundation of China (NSFC)
Publisher's version
License
File Access Level
Open
Output statusPublished
Publication dates
Online09 Nov 2022
Publication process dates
Accepted03 Nov 2022
Deposited26 Jul 2023
Grant ID1871339
1186113100
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