Numerical solution of (2+1)-dimensional nonlinear sine-Gordon equation with variable coefficients by using an efficient deep learning approach

In this article, we present an efficient neural-network-based deep learning approach, physics-informed neural networks (PINNs) with regularization technique, to resolve (2+1)-dimensional nonlinear damped and undamped sine-Gordon problem with variable coefficients. We suggest a multi-objective cost f...

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Bibliographic Details
Published in:Zeitschrift für angewandte Mathematik und Physik Vol. 76; no. 4; p. 134
Main Authors: Deresse, Alemayehu Tamirie, Dufera, Tamirat Temesgen
Format: Journal Article
Language:English
Published: Heidelberg Springer Nature B.V 01.08.2025
Subjects:
Approximation
Artificial neural networks
Boundary conditions
Boundary value problems
Cost function
Deep learning
Finite volume method
Inverse problems
Iterative methods
Machine learning
Neural networks
Numerical analysis
Parameter estimation
Partial differential equations
Physics
Regularization
ISSN:0044-2275, 1420-9039
Online Access:Get full text
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Summary:In this article, we present an efficient neural-network-based deep learning approach, physics-informed neural networks (PINNs) with regularization technique, to resolve (2+1)-dimensional nonlinear damped and undamped sine-Gordon problem with variable coefficients. We suggest a multi-objective cost function that incorporates the beginning circumstances, the governing problem’s residual, and various boundary circumstances to directly incorporate physical information of the proposed initial boundary value problem into the learning process. We employed a multiple densely connected network called feed-forward deep neural networks. To further enhance the network’s robustness and adaptability capabilities, we apply two regularization approaches to the PINNs. We point out three illustrative examples to demonstrate our suggested method’s effectiveness, validity, and practical consequences. The results showed that regularized PINNs produce better outcomes than the usual PINNs. We assess the accuracy of the model based on the relative, absolute, and training errors through tables and graphs. The findings suggested that the proposed machine learning approach PINNs is efficient and accurate and can be applied to any variable coefficient problem without requiring any linearization, perturbation, or interpolation techniques and the inclusion of appropriate regularization strategies in the PINNs can improve its performance. Therefore, to solve the variable coefficient nonlinear sine-Gordon equation and other difficult nonlinear physical issues in a range of fields, the PINNs is an appropriate programming machine learning method that is both accurate and efficient.
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ISSN:0044-2275
1420-9039
DOI:10.1007/s00033-025-02515-9