Core Concepts
A two-step approach is proposed to learn nonlinear Lagrangian reduced-order models (ROMs) of nonlinear mechanical systems directly from data, without requiring access to the full-order model operators. The method first learns a linear Lagrangian ROM via Lagrangian operator inference and then augments it with nonlinear terms learned using structure-preserving machine learning.
Abstract
The proposed Lagrangian operator inference enhanced with structure-preserving machine learning (LOpInf-SpML) method learns nonlinear Lagrangian ROMs in two steps:
Step 1 - Lagrangian Operator Inference (LOpInf):
Learns the best-fit linear Lagrangian ROM from reduced snapshot data.
Infers the linear reduced stiffness matrix and linear reduced damping matrix in a structure-preserving manner.
Step 2 - Structure-Preserving Machine Learning (SpML):
Augments the linear Lagrangian ROM with nonlinear terms learned using structure-preserving neural networks.
Learns the nonlinear components of the reduced potential energy and reduced dissipation function.
Ensures the learned ROM respects the underlying Lagrangian structure.
The method is demonstrated on three examples with increasing complexity:
Simulated data from a conservative nonlinear rod model:
The LOpInf-SpML ROM accurately captures the nonlinear dynamics and provides stable long-time predictions.
Outperforms a POD-based SpML approach in terms of accuracy and stability.
Simulated data from a nonlinear membrane model with internal damping:
The LOpInf-SpML ROM reliably captures the nonlinear characteristics, including amplitude-dependent frequency and damping.
Experimental data from a lap-joint beam structure:
The proposed method learns a predictive nonlinear ROM directly from the experimental data, without requiring access to the full-order model.
The numerical results demonstrate that the LOpInf-SpML 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.
Stats
The reduced snapshot data b
Q and the reduced time-derivative data ˙
b
Q and ¨
b
Q are used to learn the nonlinear Lagrangian ROM.