检测动力学数据中相干结构的核方法
Kernel methods for detecting coherent structures in dynamical data
摘要：
我们说明经典的基于核的降维技术和再生核希尔伯特空间（RKHS）运营商与动力系统的经验估计的特征分解之间的关系。特别是，我们表明，内核典型相关分析（CCA）可以解释的内核传输运营商，它可以通过优化马尔可夫过程（VAMP）得分的变分方法。因此，我们表明，相干集的粒子轨迹可以计算核CCA。我们通过几个例子证明了这种方法的有效性，即著名的比克利喷流、海洋漂流者数据以及具有随时间变化的势的分子动力学问题。最后，我们提出了一个简单的推广动态模式分解（DMD）称为相干模式分解（CMD）。我们的研究结果提供了一种通用的机器学习方法来计算具有客观分数的相干集，该客观分数可用于交叉验证和不同方法的比较。
英文摘要：
We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that it can be obtained by optimizing the variational approach for Markov processes (VAMP) score. As a result, we show that coherent sets of particle trajectories can be computed by kernel CCA. We demonstrate the efficiency of this approach with several examples, namely the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Finally, we propose a straightforward generalization of dynamic mode decomposition (DMD) called coherent mode decomposition (CMD). Our results provide a generic machine learning approach to the computation of coherent sets with an objective score that can be used for cross-validation and the comparison of different methods.
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