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

Aug 5, 2026

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Repeatedly solving nonlinear PDEs across varying parameters is fundamental for characterising parametric dependences of dynamical systems, yet high-fidelity simulations are often computationally prohibitive. Reduced-order modelling (ROM) enables economical low-dimensional parametric surrogates for rapid prediction of high-dimensional fields. This paper proposes a physics-data combined machine learning (PDCML) method for non-intrusive parametric ROM in small-data regimes, using POD for reduced bases and feedforward neural networks with physics-guided loss to address labelled data scarcity, tested against data-driven, physics-guided, and projection-based baselines.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages49

Reading Time2 h 3 min

AccessFree to download

Introduction

Parametric PDEs and surrogate modelling needs

Parametric reduced-order modelling and projection-based ROM

Intrusive vs non-intrusive ROM and AI connection

Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes Jinlong Fua , Dunhui Xiaob,􀀃, Rui Fua , Chenfeng Lia,c , Chuanhua Zhua , Rossella Arcuccid , Ionel M. Navone aZienkiewicz Centre for Computational Engineering, Faculty of Science and Engineering, Swansea University, Swansea SA1 8EN, UK b School of Mathematical Sciences, Tongji University, Shanghai, 200092, P.R. China c Energy Safety Research Institute, Faculty of Science and Engineering, Swansea University, Swansea SA1 8EN, UK d Department of Earth Science and Engineering, Imperial College London, London SW7 2BP, UK e Department of Scientiﬁc Computing, Florida State University, Tallahassee, FL, 32306 -4120, USA 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) providesan 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 optimization 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 optimization objective and keeping the data loss asthe 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 generalization capability and small data requirement. Keywords: Physics-data combination; Model order reduction; Feedforward neural network; Nonlinear dynamics; Non-intrusive; Small data 1. Introduction In modern engineering and scientiﬁc applications, it is a remarkably common situation to repeatedly solve parametrized partial differential equations (PDEs) over a broad range of parameter values through high-ﬁdelity numerical simulations [1, 2, 3, 4] . The parameters might describe material properties, geometric characteristics, initial conditions, boundary settings and system conﬁgurations in various disciplines, such as uncertainty quantiﬁcation, sensitive analysis, data assimilation, structure design, weather forecasting and pore-scale modelling [5, 6] . However, complete reliance on high-ﬁdelity numerical simulation to explore relevant problems usually brings a prohibitive computation burden, and even super-computing platforms can be overstretched to deal with large-scale and complicated problems. Besides, fast and nearly real-time simulations are also urgent demands in many industrial applications [7], such as online dynamic system control, structural health monitoring, online advanced manufacturing

FAQs

What is the core idea of the proposed PDCML method?

The method combines POD-based reduced-order modelling with a feedforward neural network that learns mappings from parameters to POD coefficients, while embedding governing physical rules into the training loss to reduce sensitivity to labelled-data scarcity.

How does the approach address small labelled data regimes?

It introduces a physics-guided training scheme where physics constraints are embedded into the loss function, using dynamic weighting to keep the physics term as the primary optimization objective while treating data loss as a constraint.

How is the method evaluated and what does it compare against?

It is tested on multiple nonlinear problems with different numbers of variables, and compared with data-driven ROM, physics-guided ROM, and traditional projection-based ROM methods to assess accuracy, generalisation, and data requirements.

Aug 5, 2026

Upload by Jiven