Dimitris Giannakis, Courant Institute
Data-driven approaches for spectral decomposition of ergodic dynamical systems
We discuss techniques for approximating the spectra of Koopman operators governing the evolution of observables in ergodic dynamical systems. These methods are based on representations of Koopman operators in bases for appropriate Hilbert spaces of observables learned from time-ordered measurements of the system using kernel algorithms for machine learning. We establish spectral convergence results for the point spectrum, and present regularization approaches applicable to systems with continuous spectra. We illustrate this framework with applications to toy dynamical systems and climate data.