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

Peng Zhi, Yuching Wu (), Cheng Qi, Tao Zhu, Xiao Wu and Hongyu Wu
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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

Mathematics, 2023, vol. 11, issue 12, 1-16

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.

Keywords: surrogate model; convolutional neural network; physics-informed neural networks; elliptic PDE; FEM (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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