While data-driven model reduction techniques are well-established for linearizable mechanical systems, general approaches to reducing non-linearizable systems with multiple coexisting steady states have been unavailable. In this paper, we review such a data-driven nonlinear model reduction methodology based on spectral submanifolds (SSMs). As input, this approach takes observations of unforced nonlinear oscillations to construct normal forms of the dynamics reduced to very low dimensional invariant manifolds. These normal forms capture amplitude-dependent properties and are accurate enough to provide predictions for non-linearizable system response under the additions of external forcing. We illustrate these results on examples from structural vibrations, featuring both synthetic and experimental data.

翻译：虽然数据驱动的减少模型技术对于可线性机械系统是十分成熟的,但对于减少具有多重共存稳定状态的不可线性系统,却缺乏一般办法;在本文件中,我们审查了这种基于光谱次元的以数据驱动的非线性减少模型方法。作为投入,这一方法对非非线性非线性振动进行了观测,以构建正常形式的动力衰减为极低的天体不动元体。这些正常形式捕捉了依赖振动的特性,并且足够准确,足以预测在外力作用增加的情况下无法线性系统的反应。我们用结构振动的例子来说明这些结果,其中既有合成数据,也有实验数据。