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A Stabilized Physics Informed Neural Networks Method for Wave Equations

A Stabilized Physics Informed Neural Networks Method for Wave Equations

Year:    2024

Author:    Yuling Jiao, Yuhui Liu, Jerry Zhijian Yang, Cheng Yuan

Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 4 : pp. 1100–1127

Abstract

In this article, we propose a novel stabilized physics informed neural networks method (SPINNs) for solving wave equations. In general, this method not only demonstrates theoretical convergence but also exhibits higher efficiency compared to the original PINNs. By replacing the $L^2$ norm with $H^1$ norm in the learning of initial condition and boundary condition, we theoretically proved that the error of solution can be upper bounded by the risk in SPINNs. Based on this, we decompose the error of SPINNs into approximation error, statistical error and optimization error. Furthermore, by applying the approximating theory of $ReLU^3$ networks and the learning theory on Rademacher complexity, covering number and pseudo-dimension of neural networks, we present a systematical non-asymptotic convergence analysis on our method, which shows that the error of SPINNs can be well controlled if the number of training samples, depth and width of the deep neural networks have been appropriately chosen. Two illustrative numerical examples on 1-dimensional and 2-dimensional wave equations demonstrate that SPINNs can achieve a faster and better convergence than classical PINNs method.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.OA-2024-0044

Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 4 : pp. 1100–1127

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    28

Keywords:    PINNs $ReLU^3$ neural network wave equations error analysis.

Author Details

Yuling Jiao

Yuhui Liu

Jerry Zhijian Yang

Cheng Yuan