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

TL;DR

The paper addresses the challenge of learning nonlinear reduced-order models for mechanical systems while preserving the underlying Lagrangian structure, using data-only approaches. It introduces LOpInf-SpML, a two-step method that first learns linear reduced operators via Lagrangian operator inference and then augments them with structure-preserving neural networks to model nonlinear reduced terms in the reduced Lagrangian and dissipation. The approach is validated on three increasingly complex cases—a conservative rod, a nonlinear membrane with damping, and experimental lap-joint data—showing stable, accurate long-time predictions, energy-bounded behavior, and good generalization beyond training data. The work demonstrates that combining physics-informed priors with expressive but structured neural networks yields superior nonintrusive ROMs with potential for broad engineering impact where full-order models are unavailable or costly to simulate.

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 structure-preserving 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, 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 and a 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.

Figures (13)

• Figure 1: Architecture for the LOpInf-SpML method. Starting from the FOM snapshot data $\mathbf Q$ in \ref{['eq:snapshot']}, the reduced snapshot data $\mathbf{\widehat{Q}}$ in \ref{['eq:qhat']} is obtained via projections onto the POD basis matrix $\mathbf{V}_r$. Step 1 learns the linear LOpInf ROM \ref{['eq:lopinf_rom']} from the reduced snapshot data and then step 2 uses the linear LOpInf ROM as a prior to learn the nonlinear reduced operators in the LOpInf-SpML ROM \ref{['eq:rom_form_general']} from the reduced snapshot data.
• Figure 2: Conservative rod model. The nonlinearities are localized within the region $s \in (s_{1}, s_{2})$, while the remainder of the rod is linear.
• Figure 3: Conservative rod model. (a) The LOpInf-SpML ROM correctly predicts $q_1(t)$ trajectory $100\%$ outside the training time interval whereas the POD-SpML ROM provides inaccurate predictions after $t=12$. (b) The LOpInf- SpML ROM exhibits bounded energy error behavior with an energy error of approximately $10^{-4}$. The energy error for the POD-SpML ROM, on the other hand, slowly grows with time. The magenta line in both plots indicates the end of the training time interval at $T_{\text{train}}=8$.
• Figure 4: Conservative rod model. An accurate phase space portrait obtained using the LOpInf-SpML ROM demonstrates that the proposed approach has learned the underlying nonlinear dynamics whereas the POD-SpML ROM fails to capture the qualitative behavior of the FOM in phase space.
• Figure 5: Conservative rod model. The LOpInf-SpML ROM based on the two-step learning approach starts at a significantly lower validation loss value and achieves the lowest error achieved by the POD-SpML ROM in $100\times$fewer epochs. The horizontal magenta line indicates the lowest validation error achieved by the POD-SpML ROM.