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.

翻译：复杂机械系统常因能量耗散机制、材料本构关系或几何/连接力学中的非线性而呈现强非线性行为。这些系统的数值建模会产生具有基础Lagrangian结构的非线性全阶模型。本文提出一种结合保结构机器学习的Lagrangian算子推断方法，用于学习非线性机械系统的非线性降阶模型（ROMs）。该两阶段方法首先通过Lagrangian算子推断学习最优线性Lagrangian ROM，随后提出保结构机器学习方法以学习降阶空间中的非线性特征。与现有需要已知非线性项数学形式的算子推断方法不同，本方法可纯粹从数据中学习保结构非线性ROM。从机器学习视角看，该方法通过提供先验知识加速保结构神经网络的训练，并通过在降阶空间中操作降低网络训练计算成本。该方法首先通过两个模拟算例验证：保守非线性杆模型和具有非线性内阻尼的二维非线性薄膜。最后在含搭接梁结构数字图像相关实验数据上验证，学习得到的预测模型能准确捕捉振幅依赖的频率和阻尼特性。数值结果表明，该方法可生成具有有界能量误差、可靠捕捉非线性特征且能在训练数据范围外提供精确长期预测的泛化非线性ROM。