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

Author

Listed:
  • Xi’an Li

    (Ceyear Technology Co., Ltd., Qingdao 266000, China)

  • Jinran Wu

    (School of Mathematical Sciences, Queensland University of Technology, Brisbane 4001, Australia
    The Institute for Learning Sciences and Teacher Education, Australian Catholic University, Brisbane 4000, Australia)

  • Lei Zhang

    (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
    Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
    MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China)

  • Xin Tai

    (Ceyear Technology Co., Ltd., Qingdao 266000, China)

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.

Suggested Citation

  • Xi’an Li & Jinran Wu & Lei Zhang & Xin Tai, 2022. "Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme," Mathematics, MDPI, vol. 10(22), pages 1-16, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4186-:d:967347
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    References listed on IDEAS

    as
    1. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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