  The Koopman operator provides a powerful framework for data-driven analysis
of dynamical systems. In the last few years, a wealth of numerical methods
providing finite-dimensional approximations of the operator have been proposed
(e.g. extended dynamic mode decomposition (EDMD) and its variants). While
convergence results for EDMD require an infinite number of dictionary elements,
recent studies have shown that only few dictionary elements can yield an
efficient approximation of the Koopman operator, provided that they are
well-chosen through a proper training process. However, this training process
typically relies on nonlinear optimization techniques. In this paper, we
propose two novel methods based on a reservoir computer to train the
dictionary. These methods rely solely on linear convex optimization. We
illustrate the efficiency of the method with several numerical examples in the
context of data reconstruction, prediction, and computation of the Koopman
operator spectrum. These results pave the way to the use of the reservoir
computer in the Koopman operator framework.
