Estimating Koopman operators for nonlinear dynamical systems: A nonparametric approach

The Koopman operator provides a linear description of non-linear systems exploiting an embedding into an infinite dimensional space. Dynamic Mode Decomposition and Extended Dynamic Mode Decomposition are amongst the most popular finite dimensional approximations of the Koopman Operator. In this paper we capture their core essence as a dual version of the same problem, embedding them into the Kernel framework. To do so, we leverage the RKHS as a suitable space for learning the Koopman dynamics. Learning from finite length data automatically provides a finite dimensional approximation induced by data. Simulations and comparison with standard procedures are included.