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Stiff-PDEs and Physics-Informed Neural Networks

Prakhar Sharma Orcid Logo, Llion Evans Orcid Logo, Michelle Tindall, Perumal Nithiarasu Orcid Logo

Archives of Computational Methods in Engineering, Volume: 30, Issue: 5, Pages: 2929 - 2958

Swansea University Authors: Prakhar Sharma Orcid Logo, Llion Evans Orcid Logo, Perumal Nithiarasu Orcid Logo

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DOI (Published version): 10.1007/s11831-023-09890-4

Abstract

In recent years, Physics-Informed Neural Networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problemswith conflicting boundary conditions at adjacent edges and corners. These problems have discontin...

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Published in: Archives of Computational Methods in Engineering
ISSN: 1134-3060 1886-1784
Published: Springer Science and Business Media LLC 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa62396
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PINNs still experience issues solving 3D problems, especially, problemswith conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a numberof PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation.</abstract><type>Journal Article</type><journal>Archives of Computational Methods in Engineering</journal><volume>30</volume><journalNumber>5</journalNumber><paginationStart>2929</paginationStart><paginationEnd>2958</paginationEnd><publisher>Springer Science and Business Media LLC</publisher><placeOfPublication/><isbnPrint/><isbnElectronic/><issnPrint>1134-3060</issnPrint><issnElectronic>1886-1784</issnElectronic><keywords>Physics-informed neural networks, stiff-PDEs, Parametric PDEs, thermal problems</keywords><publishedDay>1</publishedDay><publishedMonth>6</publishedMonth><publishedYear>2023</publishedYear><publishedDate>2023-06-01</publishedDate><doi>10.1007/s11831-023-09890-4</doi><url>http://dx.doi.org/10.1007/s11831-023-09890-4</url><notes/><college>COLLEGE NANME</college><department>Science and Engineering - Faculty</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>FGSEN</DepartmentCode><institution>Swansea University</institution><apcterm>SU Library paid the OA fee (TA Institutional Deal)</apcterm><funders>This work is funded by the United Kingdom Atomic Energy Authority (UKAEA) and the Engineering and Physical Sciences Research Council (EPSRC) under the Grant Agreement Numbers EP/T517987/1 and EP/R012091/1. We acknowledge the support of Supercomputing Wales and AccelerateAI projects, which is part-funded by the European Regional Development Fund (ERDF) via the Welsh Government for giving us access to NVIDIA A100 40 GB GPUs for batch training. 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spelling v2 62396 2023-01-23 Stiff-PDEs and Physics-Informed Neural Networks c940112620a47fad0bab66de278a47b5 0000-0002-7635-1857 Prakhar Sharma Prakhar Sharma true false 74dc5084c47484922a6e0135ebcb9402 0000-0002-4964-4187 Llion Evans Llion Evans true false 3b28bf59358fc2b9bd9a46897dbfc92d 0000-0002-4901-2980 Perumal Nithiarasu Perumal Nithiarasu true false 2023-01-23 FGSEN In recent years, Physics-Informed Neural Networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problemswith conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a numberof PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation. Journal Article Archives of Computational Methods in Engineering 30 5 2929 2958 Springer Science and Business Media LLC 1134-3060 1886-1784 Physics-informed neural networks, stiff-PDEs, Parametric PDEs, thermal problems 1 6 2023 2023-06-01 10.1007/s11831-023-09890-4 http://dx.doi.org/10.1007/s11831-023-09890-4 COLLEGE NANME Science and Engineering - Faculty COLLEGE CODE FGSEN Swansea University SU Library paid the OA fee (TA Institutional Deal) This work is funded by the United Kingdom Atomic Energy Authority (UKAEA) and the Engineering and Physical Sciences Research Council (EPSRC) under the Grant Agreement Numbers EP/T517987/1 and EP/R012091/1. We acknowledge the support of Supercomputing Wales and AccelerateAI projects, which is part-funded by the European Regional Development Fund (ERDF) via the Welsh Government for giving us access to NVIDIA A100 40 GB GPUs for batch training. We also acknowledge the support of NVIDIA for donating us a NVIDIA RTX A5000 24 GB for local testing. 2023-06-27T16:49:39.4968799 2023-01-23T10:22:00.7217233 Faculty of Science and Engineering School of Engineering and Applied Sciences - Biomedical Engineering Prakhar Sharma 0000-0002-7635-1857 1 Llion Evans 0000-0002-4964-4187 2 Michelle Tindall 3 Perumal Nithiarasu 0000-0002-4901-2980 4 62396__27891__c17e9454dd87403a97456d649f0c19b4.pdf 62396.VOR.pdf 2023-06-20T14:44:23.8717736 Output 10523059 application/pdf Version of Record true © The Author(s) 2023. Distributed under the terms of a Creative Commons Attribution 4.0 License (CC BY 4.0). true eng https://creativecommons.org/licenses/by/4.0/
title Stiff-PDEs and Physics-Informed Neural Networks
spellingShingle Stiff-PDEs and Physics-Informed Neural Networks
Prakhar Sharma
Llion Evans
Perumal Nithiarasu
title_short Stiff-PDEs and Physics-Informed Neural Networks
title_full Stiff-PDEs and Physics-Informed Neural Networks
title_fullStr Stiff-PDEs and Physics-Informed Neural Networks
title_full_unstemmed Stiff-PDEs and Physics-Informed Neural Networks
title_sort Stiff-PDEs and Physics-Informed Neural Networks
author_id_str_mv c940112620a47fad0bab66de278a47b5
74dc5084c47484922a6e0135ebcb9402
3b28bf59358fc2b9bd9a46897dbfc92d
author_id_fullname_str_mv c940112620a47fad0bab66de278a47b5_***_Prakhar Sharma
74dc5084c47484922a6e0135ebcb9402_***_Llion Evans
3b28bf59358fc2b9bd9a46897dbfc92d_***_Perumal Nithiarasu
author Prakhar Sharma
Llion Evans
Perumal Nithiarasu
author2 Prakhar Sharma
Llion Evans
Michelle Tindall
Perumal Nithiarasu
format Journal article
container_title Archives of Computational Methods in Engineering
container_volume 30
container_issue 5
container_start_page 2929
publishDate 2023
institution Swansea University
issn 1134-3060
1886-1784
doi_str_mv 10.1007/s11831-023-09890-4
publisher Springer Science and Business Media LLC
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Engineering and Applied Sciences - Biomedical Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Engineering and Applied Sciences - Biomedical Engineering
url http://dx.doi.org/10.1007/s11831-023-09890-4
document_store_str 1
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description In recent years, Physics-Informed Neural Networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problemswith conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a numberof PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation.
published_date 2023-06-01T16:49:34Z
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score 11.012924

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