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Nonintrusive Learning of Nonlinear Lagrangian Reduced-Order Models for Mechanical Systems


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.
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Deeper Inquiries

How can the proposed LOpInf-SpML method be extended to learn ROMs for high-dimensional mechanical systems with complex geometries and nonlinearities

The LOpInf-SpML method can be extended to learn ROMs for high-dimensional mechanical systems with complex geometries and nonlinearities by adapting the approach to handle the increased complexity. One way to extend the method is to incorporate higher-order terms in the parametrization of the nonlinear components of the reduced potential energy and dissipation functions. For high-dimensional systems, the neural networks used in the SpML part of the method can be designed with more layers and units to capture the intricate nonlinear relationships present in the system. Additionally, the reduced basis obtained from the POD can be refined to capture more detailed features of the system's dynamics. By increasing the complexity and capacity of the neural networks and refining the reduced basis, the LOpInf-SpML method can effectively learn ROMs for high-dimensional mechanical systems with complex geometries and nonlinearities.

What are the potential challenges in applying the LOpInf-SpML approach to learn ROMs for fluid-structure interaction problems or other multi-physics systems

Applying the LOpInf-SpML approach to learn ROMs for fluid-structure interaction problems or other multi-physics systems may face several challenges. One challenge is the integration of different physical domains and phenomena into the reduced model. Fluid-structure interaction involves the coupling of fluid dynamics and structural mechanics, which requires a comprehensive understanding of both domains. The LOpInf-SpML method would need to capture the interactions and nonlinearities between the fluid and structure accurately, which can be complex and computationally intensive. Another challenge is the validation and verification of the ROMs for multi-physics systems, as the accuracy and stability of the reduced models need to be rigorously tested across different operating conditions and scenarios. Additionally, the high-dimensional nature of multi-physics systems can pose challenges in terms of computational resources and training complexity for the neural networks in the SpML part of the method.

Can the structure-preserving machine learning techniques used in the LOpInf-SpML method be combined with other data-driven model reduction approaches, such as the Koopman operator framework, to learn interpretable nonlinear ROMs

The structure-preserving machine learning techniques used in the LOpInf-SpML method can be combined with other data-driven model reduction approaches, such as the Koopman operator framework, to learn interpretable nonlinear ROMs. By integrating the structure-preserving principles of the SpML methods with the Koopman operator framework, the resulting ROMs can capture the underlying geometric and conservation properties of the system while also providing insights into the system's dynamics. The Koopman operator framework can be used to analyze the system's behavior in a high-dimensional space, while the SpML techniques can help in learning the reduced dynamics in a more interpretable and structured manner. This combination can lead to ROMs that not only accurately represent the system's behavior but also provide valuable insights into the underlying physics and dynamics.
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