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Solving a class of multi-scale elliptic PDEs by Fourier-based mixed physics informed neural networks
Li, Xi-An Wu, Jinran Tai, Xin Xu, Jianhua Wang, You-Gan
Li, Xi-An
Wu, Jinran
Tai, Xin
Xu, Jianhua
Wang, You-Gan
Abstract
Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations (PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks (dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix caused by the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be publicly available at https://github.com/Blue-Giant/FMPINN.
Keywords
Multi-scale, Rough coefficient, FMPINN, Fourier feature mapping, Flux variable, Reduce order
Date
2024
Type
Journal article
Journal
Book
Volume
508
Issue
C
Page Range
1-23
Article Number
ACU Department
Institute for Positive Psychology and Education
Faculty of Education and Arts
Institute for Learning Sciences and Teacher Education (ILSTE)
Faculty of Education and Arts
Institute for Learning Sciences and Teacher Education (ILSTE)
Files
Relation URI
Event URL
Open Access Status
Open access
License
CC BY 4.0
File Access
Open
Notes
© 2024 The Author(s). Published by Elsevier Inc.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).