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DWNN: Deep Wavelet Neural Network for Solving Partial Differential Equations

Author

Listed:
  • Ying Li

    (School of Computer Engineering and Science, Shanghai University, Shanghai 200444, China)

  • Longxiang Xu

    (School of Computer Engineering and Science, Shanghai University, Shanghai 200444, China)

  • Shihui Ying

    (Department of Mathematics, School of Science, Shanghai University, Shanghai 200444, China)

Abstract

In this paper, we propose a deep wavelet neural network (DWNN) model to approximate the natural phenomena that are described by some classical PDEs. Concretely, we introduce wavelets to deep architecture to obtain a fine feature description and extraction. That is, we constructs a wavelet expansion layer based on a family of vanishing momentum wavelets. Second, the Gaussian error function is considered as the activation function owing to its fast convergence rate and zero-centered output. Third, we design the cost function by considering the residual of governing equation, the initial/boundary conditions and an adjustable residual term of observations. The last term is added to deal with the shock wave problems and interface problems, which is conducive to rectify the model. Finally, a variety of numerical experiments are carried out to demonstrate the effectiveness of the proposed approach. The numerical results validate that our proposed method is more accurate than the state-of-the-art approach.

Suggested Citation

  • Ying Li & Longxiang Xu & Shihui Ying, 2022. "DWNN: Deep Wavelet Neural Network for Solving Partial Differential Equations," Mathematics, MDPI, vol. 10(12), pages 1-35, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:1976-:d:834066
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    References listed on IDEAS

    as
    1. Yang, Xiaojia & Ge, Yongbin & Zhang, Lin, 2019. "A class of high-order compact difference schemes for solving the Burgers’ equations," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 394-417.
    2. 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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    Cited by:

    1. Zebin Xing & Heng Cheng & Jing Cheng, 2023. "Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems," Mathematics, MDPI, vol. 11(19), pages 1-21, September.

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