  We propose an alternating optimization algorithm to the nonconvex Koopman
operator learning problem for nonlinear dynamic systems. We show that the
proposed algorithm will converge to a critical point with rate $O(1/T)$ and
$O(\frac{1}{\log T})$ for the constant and diminishing learning rates,
respectively, under some mild conditions. To cope with the high dimensional
nonlinear dynamical systems, we present the first-ever distributed Koopman
operator learning algorithm. We show that the distributed Koopman operator
learning has the same convergence properties as the centralized Koopman
operator learning, in the absence of optimal tracker, so long as the basis
functions satisfy a set of state-based decomposition conditions. Numerical
experiments are provided to complement our theoretical results.
