Adaptive Deep Learning Approximation for Allen-Cahn Equation

Huiying Xu; Jie Chen; Fei Ma

doi:10.1007/978-3-031-08760-8_23

Adaptive Deep Learning Approximation for Allen-Cahn Equation

Huiying Xu, Jie Chen^*, Fei Ma

^*Corresponding author for this work

Xi'an Jiaotong-Liverpool University

Research output: Chapter in Book or Report/Conference proceeding › Conference Proceeding › peer-review

3 Citations (Scopus)

Abstract

Solving general non-linear partial differential equations (PDE) precisely and efficiently has been a long-lasting challenge in the field of scientific computing. Based on the deep learning framework for solving non-linear PDEs physics-informed neural networks (PINN), we introduce an adaptive collocation strategy into the PINN method to improve the effectiveness and robustness of this algorithm when selecting the initial data to be trained. Instead of merely training the neural network once, multi-step discrete time models are considered when predicting the long time behaviour of solutions of the Allen-Cahn equation. Numerical results concerning solutions of the Allen-Cahn equation are presented, which demonstrate that this approach can improve the robustness of original neural networks approximation.

Original language	English
Title of host publication	Computational Science - ICCS 2022, 22nd International Conference, Proceedings
Editors	Derek Groen, Clélia de Mulatier, Valeria V. Krzhizhanovskaya, Peter M.A. Sloot, Maciej Paszynski, Jack J. Dongarra
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	271-283
Number of pages	13
ISBN (Print)	9783031087592
DOIs	https://doi.org/10.1007/978-3-031-08760-8_23
Publication status	Published - 2022
Event	22nd Annual International Conference on Computational Science, ICCS 2022 - London, United Kingdom Duration: 21 Jun 2022 → 23 Jun 2022

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	13353 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	22nd Annual International Conference on Computational Science, ICCS 2022
Country/Territory	United Kingdom
City	London
Period	21/06/22 → 23/06/22

Keywords

Adaptive collocation
Allen-Cahn equation
Deep learning
Discrete time models
Physics informed neural networks

Access to Document

10.1007/978-3-031-08760-8_23

Cite this

Xu, H., Chen, J., & Ma, F. (2022). Adaptive Deep Learning Approximation for Allen-Cahn Equation. In D. Groen, C. de Mulatier, V. V. Krzhizhanovskaya, P. M. A. Sloot, M. Paszynski, & J. J. Dongarra (Eds.), Computational Science - ICCS 2022, 22nd International Conference, Proceedings (pp. 271-283). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 13353 LNCS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-031-08760-8_23

Xu, Huiying ; Chen, Jie ; Ma, Fei. / Adaptive Deep Learning Approximation for Allen-Cahn Equation. Computational Science - ICCS 2022, 22nd International Conference, Proceedings. editor / Derek Groen ; Clélia de Mulatier ; Valeria V. Krzhizhanovskaya ; Peter M.A. Sloot ; Maciej Paszynski ; Jack J. Dongarra. Springer Science and Business Media Deutschland GmbH, 2022. pp. 271-283 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "Solving general non-linear partial differential equations (PDE) precisely and efficiently has been a long-lasting challenge in the field of scientific computing. Based on the deep learning framework for solving non-linear PDEs physics-informed neural networks (PINN), we introduce an adaptive collocation strategy into the PINN method to improve the effectiveness and robustness of this algorithm when selecting the initial data to be trained. Instead of merely training the neural network once, multi-step discrete time models are considered when predicting the long time behaviour of solutions of the Allen-Cahn equation. Numerical results concerning solutions of the Allen-Cahn equation are presented, which demonstrate that this approach can improve the robustness of original neural networks approximation.",

keywords = "Adaptive collocation, Allen-Cahn equation, Deep learning, Discrete time models, Physics informed neural networks",

author = "Huiying Xu and Jie Chen and Fei Ma",

note = "Funding Information: J. Chen—This work is partially supported by Key Program Special Fund in XJTLU (KSF-E-50, KSF-E-21) and XJTLU Research Development Funding (RDF-19-01-15). Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.; 22nd Annual International Conference on Computational Science, ICCS 2022 ; Conference date: 21-06-2022 Through 23-06-2022",

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Xu, H, Chen, J & Ma, F 2022, Adaptive Deep Learning Approximation for Allen-Cahn Equation. in D Groen, C de Mulatier, VV Krzhizhanovskaya, PMA Sloot, M Paszynski & JJ Dongarra (eds), Computational Science - ICCS 2022, 22nd International Conference, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 13353 LNCS, Springer Science and Business Media Deutschland GmbH, pp. 271-283, 22nd Annual International Conference on Computational Science, ICCS 2022, London, United Kingdom, 21/06/22. https://doi.org/10.1007/978-3-031-08760-8_23

Adaptive Deep Learning Approximation for Allen-Cahn Equation. / Xu, Huiying; Chen, Jie ; Ma, Fei.
Computational Science - ICCS 2022, 22nd International Conference, Proceedings. ed. / Derek Groen; Clélia de Mulatier; Valeria V. Krzhizhanovskaya; Peter M.A. Sloot; Maciej Paszynski; Jack J. Dongarra. Springer Science and Business Media Deutschland GmbH, 2022. p. 271-283 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 13353 LNCS).

Research output: Chapter in Book or Report/Conference proceeding › Conference Proceeding › peer-review

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N1 - Funding Information: J. Chen—This work is partially supported by Key Program Special Fund in XJTLU (KSF-E-50, KSF-E-21) and XJTLU Research Development Funding (RDF-19-01-15). Publisher Copyright: © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

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N2 - Solving general non-linear partial differential equations (PDE) precisely and efficiently has been a long-lasting challenge in the field of scientific computing. Based on the deep learning framework for solving non-linear PDEs physics-informed neural networks (PINN), we introduce an adaptive collocation strategy into the PINN method to improve the effectiveness and robustness of this algorithm when selecting the initial data to be trained. Instead of merely training the neural network once, multi-step discrete time models are considered when predicting the long time behaviour of solutions of the Allen-Cahn equation. Numerical results concerning solutions of the Allen-Cahn equation are presented, which demonstrate that this approach can improve the robustness of original neural networks approximation.

AB - Solving general non-linear partial differential equations (PDE) precisely and efficiently has been a long-lasting challenge in the field of scientific computing. Based on the deep learning framework for solving non-linear PDEs physics-informed neural networks (PINN), we introduce an adaptive collocation strategy into the PINN method to improve the effectiveness and robustness of this algorithm when selecting the initial data to be trained. Instead of merely training the neural network once, multi-step discrete time models are considered when predicting the long time behaviour of solutions of the Allen-Cahn equation. Numerical results concerning solutions of the Allen-Cahn equation are presented, which demonstrate that this approach can improve the robustness of original neural networks approximation.

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T2 - 22nd Annual International Conference on Computational Science, ICCS 2022

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Xu H, Chen J , Ma F. Adaptive Deep Learning Approximation for Allen-Cahn Equation. In Groen D, de Mulatier C, Krzhizhanovskaya VV, Sloot PMA, Paszynski M, Dongarra JJ, editors, Computational Science - ICCS 2022, 22nd International Conference, Proceedings. Springer Science and Business Media Deutschland GmbH. 2022. p. 271-283. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-031-08760-8_23