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Surrogate-Based Physics-Informed Neural Networks for Elliptic Partial Differential Equations

Author

Listed:
  • Peng Zhi

    (College of Civil Engineering, Tongji University, Shanghai 200092, China)

  • Yuching Wu

    (College of Civil Engineering, Tongji University, Shanghai 200092, China)

  • Cheng Qi

    (College of Civil Engineering, Tongji University, Shanghai 200092, China)

  • Tao Zhu

    (College of Civil Engineering, Tongji University, Shanghai 200092, China)

  • Xiao Wu

    (College of Civil Engineering, Tongji University, Shanghai 200092, China)

  • Hongyu Wu

    (College of Civil Engineering, Tongji University, Shanghai 200092, China)

Abstract

The purpose of this study is to investigate the role that a deep learning approach could play in computational mechanics. In this paper, a convolutional neural network technique based on modified loss function is proposed as a surrogate of the finite element method (FEM). Several surrogate-based physics-informed neural networks (PINNs) are developed to solve representative boundary value problems based on elliptic partial differential equations (PDEs). According to the authors’ knowledge, the proposed method has been applied for the first time to solve boundary value problems with elliptic partial differential equations as the governing equations. The results of the proposed surrogate-based approach are in good agreement with those of the conventional FEM. It is found that modification of the loss function could improve the prediction accuracy of the neural network. It is demonstrated that to some extent, the deep learning approach could replace the conventional numerical method as a significant surrogate model.

Suggested Citation

  • Peng Zhi & Yuching Wu & Cheng Qi & Tao Zhu & Xiao Wu & Hongyu Wu, 2023. "Surrogate-Based Physics-Informed Neural Networks for Elliptic Partial Differential Equations," Mathematics, MDPI, vol. 11(12), pages 1-16, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2723-:d:1172133
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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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