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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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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
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa62140
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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 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.
Keywords: Physics-data combination; Model order reduction; Feedforward neural network; Nonlinear dynamics; Non-intrusive; Small data
College: Faculty of Science and Engineering
Funders: The authors would like to acknowledge the support of EPSRC, United Kingdom grant: PURIFY ( EP/V000756/1).
Start Page: 115771