System identification using the Koopman operator: some quantitative considerations

Abstract

The Koopman operator is an important object in the study of dynamical systems. For a given nonlinear dynamical system on a given domain, it defines a related linear transport equation on that same domain.Thus, it turns a finite-dimensional, nonlinear system, into an infinite-dimensional, linear system. This newfound linearity can then be leveraged to handle system identification: given a system for which the dynamics are unknown, is it possible to extract a model of the dynamics from observations of the system? At what cost? More precisely, given a system assumed to be governed by an autonomous differential equation, how can one recover the vector field of that differential equation from trajectory data? Can this be used for control applications?In this talk we will illustrate how the Koopman operator can help answer these questions with the so-called extended Dynamic Mode Decomposition algorithm (introduced by Williams, Kevrekidis & Rowley). We will show some quantitative error estimates and simulations. In particular, the quantitative estimates illustrate clearly the difficulties and challenges of this method, such as the curse of dimensionality.