Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

TL;DR

This work tackles the challenge of obtaining accurate, long-time stable reduced-order models for large-scale Lagrangian systems without access to full-order operators. It introduces L-OpInf, a constrained, nonintrusive framework that learns reduced Lagrangian ROMs by enforcing SPD structure on reduced operators, ensuring energy behavior mirrors the FOM. Through three challenging problems—an Euler–Bernoulli beam, a nonlinear sine–Gordon wave equation, and a soft-robot fishtail with nearly 780k DOFs—the method demonstrates accurate predictions far outside the training window, energy-bounded dynamics, and robustness to unseen inputs. The approach significantly lowers computational cost while preserving physical fidelity, enabling real-time simulation and control of complex, large-scale dynamical systems.

Abstract

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler-Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler-Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

Figures (17)

• Figure 1: Schematic overview of the types of Lagrangian FOMs considered in this work and the respective application problems (dark grey).
• Figure 2: Euler-Bernoulli beam: A schematic showing transverse vibrations in response to a nonzero initial condition. The beam is simply supported at both ends which allows for rotation but not for vertical displacement.
• Figure 3: Euler-Bernoulli beam: Even though plot (a) shows low state approximation error for second-order operator inference in the training data regime, the corresponding FOM energy error \ref{['eq:fom_energy_error']} behavior in plot (b) reveals that the standard second-order operator inference violates the underlying Lagrangian structure which leads to unbounded growth of the energy error outside the training data. The vertical black line in plot (b) indicates end of training time interval $[0, 0.03]$ s, which is a small fraction of the full simulation time shown in (b). While this training data is enough to learn the model accurately (see (a)), it still does not produce a conservative Lagrangian system.
• Figure 4: Euler-Bernoulli beam: Computational cost of solving the constrained optimization problem \ref{['eq:lopinf']} for ROMs of reduced dimension $2r$ and different sizes of the training data, $K$.
• Figure 5: Euler-Bernoulli beam: L-OpInf ROMs achieve lower state error than intrusive Lagrangian ROMs in both training and test interval, yet in the testing interval the state errors level off after $r\geq 12$.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3