Kernel methods for detecting coherent structures in dynamical data.

Area of Science:

• Dynamical systems analysis
• Machine learning
• Data science

Background:

• Classical dimensionality reduction techniques are essential for understanding complex systems.
• Reproducing kernel Hilbert space operators are key to analyzing dynamical systems.
• Kernel canonical correlation analysis (CCA) is a powerful tool for feature extraction.

Purpose of the Study:

• To establish connections between kernel-based dimensionality reduction and dynamical system operators.
• To demonstrate that kernel CCA can compute coherent sets of particle trajectories.
• To introduce a generalized dynamic mode decomposition method called coherent mode decomposition.

Main Methods:

• Kernel-based dimensionality reduction
• Eigendecomposition of empirical estimates
• Kernel canonical correlation analysis (CCA)
• Variational approach for Markov processes score optimization
• Coherent mode decomposition

Main Results:

• Kernel CCA is interpretable via kernel transfer operators and Markov processes.
• Coherent sets of particle trajectories can be effectively computed using kernel CCA.
• Demonstrated efficiency on Bickley jet, ocean drifter, and molecular dynamics data.

Conclusions:

• The proposed kernel CCA approach provides a generic machine learning framework for computing coherent sets.
• The objective score facilitates cross-validation and method comparison.
• Coherent mode decomposition offers a novel generalization of dynamic mode decomposition.