采用保结构机器学习增强的 Lagrangian 算子推断，用于机械系统的非侵入式模型降阶
Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems
AI总结：
针对具有 Lagrangian 结构的非线性机械系统，提出数据驱动的 Lagrangian 算子推断与保结构神经网络结合方法，构建可泛化、能量误差有界且长时预测准确的非线性 ROM。
摘要：
复杂机械系统往往表现出强非线性行为，其原因在于能量耗散机制、材料本构关系或几何/连接力学中存在非线性。对这些系统进行数值建模会产生非线性全阶模型，而这些模型具有潜在的 Lagrangian 结构。本文提出一种采用保结构机器学习增强的 Lagrangian 算子推断方法，以学习非线性机械系统的非线性降阶模型（ROM）。该两步方法首先通过 Lagrangian 算子推断学习最佳拟合的线性 Lagrangian ROM，随后提出一种保结构机器学习方法，用于在降阶空间中学习非线性。与现有需要获知非线性项数学形式的算子推断方法不同，所提出的方法能够完全基于数据学习保结构非线性 ROM。从机器学习角度看，该方法通过提供有信息的先验来加速保结构神经网络的训练，并通过在降阶空间上运算来降低网络训练的计算成本。该方法首先在两个仿真算例上进行演示：一个保守非线性杆模型，以及一个具有非线性内阻尼的二维非线性膜。最后，在一个实验数据集上演示该方法；该数据集由从搭接接头梁结构采集的数字图像相关测量结果构成，并由此学习得到一个预测模型，能够准确捕捉依赖于振幅的频率和阻尼特性。数值结果表明，所提出的方法能够产生可泛化的非线性 ROM，这些模型表现出有界的能量误差，可靠捕捉非线性特征，并在训练数据区间之外提供准确的长时间预测。
英文摘要：
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.
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