Reproducing kernel Hilbert space compactification of unitary evolution groups

Abstract

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. It is based on an approximation of the generator of the system by a compact operator $W_\tau$ on a reproducing kernel Hilbert space (RKHS). A key tool is that $W_\tau$ is skew-adjoint , and thus can be characterized by a unique projection-valued measure, discrete by compactness, and an associated orthonormal basis of eigenfunctions. These eigenfunctions can be ordered in terms of a Dirichlet energy on the RKHS, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, the regularized generator has a well-defined Borel functional calculus allowing the construction of a unitary evolution group $\{ e^{t W_\tau} \}_{t\in\mathbb{R}}$ on the RKHS, which approximates the unitary Koopman evolution group of the original system. We establish convergence results for the spectrum and Borel functional calculus of the regularized generator to those of the original system in the limit $\tau \to 0^+$. Convergence results are also established for a data-driven formulation, where these operators are approximated using finite-rank operators obtained from observed time series. An advantage of working with an RKHS structure is that one can perform pointwise evaluation and interpolation through bounded linear operators, which is not possible in $L^p$ spaces. This enables the out of sample evaluation of data-approximated eigenfunctions, as well as data-driven forecasts initialized with pointwise initial data (as opposed to probability densities in $L^p$). The pattern extraction and prediction framework is numerically applied to three instances of ergodic dynamical systems with atomic and continuous spectra.