CMX Lunch

It is a remarkable fact, realized in the work of Koopman in the 1930s, that a general deterministic dynamical system can be characterized through its action on spaces of observables (functions of the state) through intrinsically linear evolution operators, acting by composition with the flow map. In the setting of measure-preserving, ergodic dynamics, these operators form a unitary group, whose spectral properties are useful for coherent pattern extraction and prediction of observables, among many applications. In this talk, we will discuss methods for data-driven approximation of the spectra of such unitary groups, focusing, in particular, on systems with mixing (chaotic) dynamics and continuous spectra. These methods utilize techniques from reproducing kernel Hilbert space (RKHS) theory to approximate the generator of the unitary Koopman group (an unbounded operator with complicated spectral behavior), through compact, skew-adjoint operators acting on a suitable RKHS of observables. These "compactified" generators have well-defined, purely atomic spectral measures, which are shown to converge to the spectrum of the generator in a limit of vanishing regularization parameter. In addition, the spectral measures of the compactified generators identify coherent observables under the dynamics through corresponding eigenfunctions, and have an associated functional calculus, allowing one to approximate functions of the generator. In particular, exponentiating the generator leads to an approximation of the unitary Koopman operator, which can be used to perform prediction of observables. The RKHS structure also allows stable, data-driven formulations of this framework that converge under fairly general assumptions on the system and observation modality. We illustrate this approach with applications to toy dynamical systems and examples drawn from climate dynamics.