Kernel methods for detecting coherent structures in dynamical data

Abstract

ABSTRACT We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space 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 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 called coherent mode decomposition. 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. While coherent sets of particles are common in dynamical systems, they are notoriously challenging to identify. In this article, we leverage the combination of a suite of methods designed to approximate the eigenfunctions of transfer operators with kernel embeddings in order to design an algorithm for detecting coherent structures in Langrangian data. It turns out that the resulting method is a well-known technique to analyze relationships between multidimensional variables, namely, kernel canonical correlation analysis (CCA). Our algorithm successfully identifies coherent structures in several diverse examples, including oceanic currents and a molecular dynamics problem with a moving potential. Furthermore, we show that a natural extension of our algorithm leads to a coherent mode decomposition (CMD), a counterpart to dynamic mode decomposition (DMD). I. INTRODUCTION