Massimiliano Pontil (Italian Institute of Technology and University College London)

Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces

Abstract In the last couple of years the Koopman operator framework attracted a lot of attention due to growing use of data-driven dynamical systems in science and engineering. Relevant results span fields of functional analysis, numerical algorithms, as well as machine learning. In this talk we will focus on discrete dynamical systems modelled by Markov chains and their study through the corresponding Koopman operator. While many data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We will introduce a framework to learn the Koopman operator from a finite data sample of the dynamical system. We will limit to the important case when the Markov chain admits an invariant distribution and, relying on the theory of reproducing kernel Hilbert spaces (RKHS), introduce a notion of risk from which different estimators naturally arise. Special attention will be given to the estimation of the Koopman spectral properties, and in particular its modal decomposition (KMD) that is of paramount importance in practice. With this framework we will analyse, theoretically and empirically, some existing estimators, e.g. (extended) dynamic mode decomposition (DMD), and introduce a novel reduced-rank operator regression (RRR) estimator. We will further present learning bounds for the proposed estimator, holding for both data drawn i.i.d. from the invariant distribution and for non i.i.d. data gathered from the trajectories of the dynamical system, the latter in terms of mixing coefficients.
(Joint work with V. Kostic, P. Novelli, A. Maurer, C. Ciliberto, L. Rosasco)