Kernel methods for detecting coherent structures in dynamical data
This provides a generic machine learning approach for analyzing coherent structures in dynamical systems, with applications in fields like fluid dynamics and molecular dynamics, though it appears incremental as it builds on existing kernel methods.
The paper tackles the problem of detecting coherent structures in dynamical systems by showing that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and optimized via the VAMP score, enabling computation of coherent sets from particle trajectories. It demonstrates efficiency on examples like the Bickley jet and ocean drifter data, and proposes a generalization called coherent mode decomposition (CMD).
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.