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.