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Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes

Jinlong Fu, Dunhui Xiao, Rui Fu, Chenfeng Li Orcid Logo, Chuanhua Zhu, Rossella Arcucci, Ionel M. Navon

Computer Methods in Applied Mechanics and Engineering, Volume: 404, Start page: 115771

Swansea University Author: Chenfeng Li Orcid Logo

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DOI (Published version): 10.1016/j.cma.2022.115771

Abstract

Repeatedly solving nonlinear partial differential equations with varying parameters is often an essential requirement to characterise the parametric dependences of dynamical systems. Reduced-order modelling (ROM) provides an economical way to construct low-dimensional parametric surrogates for rapid...

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Published in: Computer Methods in Applied Mechanics and Engineering
ISSN: 0045-7825
Published: Elsevier BV 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa62140
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This paper presents a physics-data combined machine learning (PDCML) method for non-intrusive parametric ROM in small-data regimes. Proper orthogonal decomposition (POD) is adopted for dimension reduction by deriving basis functions from a limited number of high-fidelity snapshots, and parametric ROM is thus transformed into establishing reliable mappings between the system parameters and the POD coefficients. To overcome labelled data scarcity, a physics-data combined ROM framework is developed to jointly integrate the physical principle and the small labelled data into feedforward neural networks (FNN) via a step-by-step training scheme. Specifically, a preliminary FNN model is firstly fitted via data-driven training, and then the governing physical rules are embedded into the loss function to improve the model interpolation and extrapolation performances through physics-guided training constrained by the labelled data. During the constrained optimisation procedure, dynamic weighting factors are used to adjust the physics-data proportion of the loss functions, aiming at continuously highlighting the physics loss as the primary optimisation objective and keeping the data loss as the constraint. This new PDCML method is tested on a series of nonlinear problems with different numbers of physical variables, and it is also compared with the data-driven ROM, the physics-guided ROM and the traditional projection-based ROM methods. 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spelling 2023-01-26T13:34:22.1134510 v2 62140 2022-12-08 Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes 82fe170d5ae2c840e538a36209e5a3ac 0000-0003-0441-211X Chenfeng Li Chenfeng Li true false 2022-12-08 CIVL Repeatedly solving nonlinear partial differential equations with varying parameters is often an essential requirement to characterise the parametric dependences of dynamical systems. Reduced-order modelling (ROM) provides an economical way to construct low-dimensional parametric surrogates for rapid predictions of high-dimensional physical fields. This paper presents a physics-data combined machine learning (PDCML) method for non-intrusive parametric ROM in small-data regimes. Proper orthogonal decomposition (POD) is adopted for dimension reduction by deriving basis functions from a limited number of high-fidelity snapshots, and parametric ROM is thus transformed into establishing reliable mappings between the system parameters and the POD coefficients. To overcome labelled data scarcity, a physics-data combined ROM framework is developed to jointly integrate the physical principle and the small labelled data into feedforward neural networks (FNN) via a step-by-step training scheme. Specifically, a preliminary FNN model is firstly fitted via data-driven training, and then the governing physical rules are embedded into the loss function to improve the model interpolation and extrapolation performances through physics-guided training constrained by the labelled data. During the constrained optimisation procedure, dynamic weighting factors are used to adjust the physics-data proportion of the loss functions, aiming at continuously highlighting the physics loss as the primary optimisation objective and keeping the data loss as the constraint. This new PDCML method is tested on a series of nonlinear problems with different numbers of physical variables, and it is also compared with the data-driven ROM, the physics-guided ROM and the traditional projection-based ROM methods. The results demonstrate that the proposed method provides a cost-effective way for non-intrusive parametric ROM via machine learning, and it possesses good characteristics of high prediction accuracy, strong generalisation capability and small data requirement. Journal Article Computer Methods in Applied Mechanics and Engineering 404 115771 Elsevier BV 0045-7825 Physics-data combination; Model order reduction; Feedforward neural network; Nonlinear dynamics; Non-intrusive; Small data 1 2 2023 2023-02-01 10.1016/j.cma.2022.115771 COLLEGE NANME Civil Engineering COLLEGE CODE CIVL Swansea University The authors would like to acknowledge the support of EPSRC, United Kingdom grant: PURIFY ( EP/V000756/1). 2023-01-26T13:34:22.1134510 2022-12-08T09:29:16.2464447 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering Jinlong Fu 1 Dunhui Xiao 2 Rui Fu 3 Chenfeng Li 0000-0003-0441-211X 4 Chuanhua Zhu 5 Rossella Arcucci 6 Ionel M. Navon 7 Under embargo Under embargo 2023-01-03T08:42:02.3069926 Output 2184526 application/pdf Accepted Manuscript true 2023-12-06T00:00:00.0000000 ©2022 All rights reserved. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND) true eng https://creativecommons.org/licenses/by-nc-nd/4.0/
title Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes
spellingShingle Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes
Chenfeng Li
title_short Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes
title_full Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes
title_fullStr Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes
title_full_unstemmed Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes
title_sort Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes
author_id_str_mv 82fe170d5ae2c840e538a36209e5a3ac
author_id_fullname_str_mv 82fe170d5ae2c840e538a36209e5a3ac_***_Chenfeng Li
author Chenfeng Li
author2 Jinlong Fu
Dunhui Xiao
Rui Fu
Chenfeng Li
Chuanhua Zhu
Rossella Arcucci
Ionel M. Navon
format Journal article
container_title Computer Methods in Applied Mechanics and Engineering
container_volume 404
container_start_page 115771
publishDate 2023
institution Swansea University
issn 0045-7825
doi_str_mv 10.1016/j.cma.2022.115771
publisher Elsevier BV
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 Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering
document_store_str 0
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description Repeatedly solving nonlinear partial differential equations with varying parameters is often an essential requirement to characterise the parametric dependences of dynamical systems. Reduced-order modelling (ROM) provides an economical way to construct low-dimensional parametric surrogates for rapid predictions of high-dimensional physical fields. This paper presents a physics-data combined machine learning (PDCML) method for non-intrusive parametric ROM in small-data regimes. Proper orthogonal decomposition (POD) is adopted for dimension reduction by deriving basis functions from a limited number of high-fidelity snapshots, and parametric ROM is thus transformed into establishing reliable mappings between the system parameters and the POD coefficients. To overcome labelled data scarcity, a physics-data combined ROM framework is developed to jointly integrate the physical principle and the small labelled data into feedforward neural networks (FNN) via a step-by-step training scheme. Specifically, a preliminary FNN model is firstly fitted via data-driven training, and then the governing physical rules are embedded into the loss function to improve the model interpolation and extrapolation performances through physics-guided training constrained by the labelled data. During the constrained optimisation procedure, dynamic weighting factors are used to adjust the physics-data proportion of the loss functions, aiming at continuously highlighting the physics loss as the primary optimisation objective and keeping the data loss as the constraint. This new PDCML method is tested on a series of nonlinear problems with different numbers of physical variables, and it is also compared with the data-driven ROM, the physics-guided ROM and the traditional projection-based ROM methods. The results demonstrate that the proposed method provides a cost-effective way for non-intrusive parametric ROM via machine learning, and it possesses good characteristics of high prediction accuracy, strong generalisation capability and small data requirement.
published_date 2023-02-01T04:21:32Z
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score 11.014067

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