Spatiotemporal Feature Extraction with Data-Driven Koopman Operators

We present a framework for feature extraction and mode decomposition of spatiotemporal data generated by ergodic dynamical systems. Unlike feature extraction techniques based on kernel operators, our approach is to construct feature maps using eigenfunctions of the Koopman group of unitary operators governing the dynamical evolution of observables and probability measures. We compute the eigenvalues and eigenfunctions of the Koopman group through a Galerkin scheme applied to time-ordered data without requiring a priori knowledge of the dynamical evolution equations. This scheme employs a data-driven set of basis functions on the state space manifold, computed through the diffusion maps algorithm and a variable-bandwidth kernel designed to enforce orthogonality with respect to the invariant measure of the dynamics. The features extracted via this approach have strong timescale separation, favorable predictability properties, and high smoothness on the state space manifold. The extracted features are also invariant under weakly restrictive changes of observation modality. We apply this scheme to a synthetic dataset featuring superimposed traveling waves in a one-dimensional periodic domain and satellite observations of organized convection in the tropical atmosphere.