Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems

Aug 4, 2026

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Complex mechanical systems exhibit strongly nonlinear behavior driven by nonlinear energy dissipation, material constitutive laws, and geometric or connectivity effects. Their numerical models form nonlinear full-order models with an underlying Lagrangian structure. This work introduces a Lagrangian operator inference framework enhanced by structure-preserving machine learning to learn nonlinear reduced-order models purely from data. A linear structure-preserving ROM is first inferred, then nonlinear terms are learned in the reduced space to enable bounded-energy errors, reliable nonlinear capture, and accurate long-time predictions beyond training regimes, validated on simulations and an experimental digital image correlation dataset.

LanguageEnglish

CategoryResearch & Report

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Pages24

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Introduction

Reduced-order models and Lagrangian structure

Operator inference and nonintrusive learning

Structure-preserving methods and limitations

Deep learning ROMs and geometric structure preservation

arXiv :2404 .05040v 1 [ cs .CE] 7 Apr 2024 Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems Harsh Sharmaa,1,∗, David A. Najera-Floresb,c,1 , Michael D. Toddb , Boris Kramera a Department of Mechanical and Aerospace Engineering, University of California San Diego, CA, United States b Department of Structural Engineering, University of California San Diego, CA, United States cATA Engineering, Inc., San Diego, CA, United States Abstract Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structurepreserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior (i.e., the linear Lagrangian ROM structure), and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model anda two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime. 1. Introduction Reduced-order models (ROMs) of nonlinear mechanical models play a key role in a variety of tasks ranging from control of soft robotics [1, 2] to design optimization of mechanical structures [3, 4] to state assessment for structural health monitoring [5, 6] . Nonlinear mechanical models in structural and mechanical engineering applications often possess an underlying Lagrangian structure. Deriving a Lagrangian ROM of these nonlinear mechanical models is of particular importance because the Lagrangian structure is intimately connected to physically interpretable quantities such as momentum and energy, or in case of fluid systems, vorticity. Structure-preserving model reduction of mechanical systems was introduced in [7] where the authors showed that performing a Galerkin projection on the Euler-Lagrange equation leads to a Lagrangian ROM. The work in [8] presented a computationally efficient, structure-preserving model reduction method for parametric Lagrangian systems with higher-order nonlinearities. Both of these structure-preserving model reduction approaches are intrusive in that they require access to full-order model (FOM) operators in order to derive nonlinear ROMs via the intrusive projection. This type of information, however, is typically unavailable when working with proprietary software, complicated legacy code, or experimental data. Thus, nonintrusive methods that can learn ROMs directly from simulated or experimental data have become increasingly popular. The operat

FAQs

Why is structure preservation important in this context?

Preserving the underlying geometric/Lagrangian structure improves physical interpretability and helps produce accurate predictions, especially for long-time behavior outside the training data regime.

Aug 4, 2026

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