Intrinsic and Extrinsic Approximation of Koopman Operators over Manifolds

Abstract: This paper derives rates of convergence of certain approximations of the Koopman operators that are associated with discrete, deterministic, continuous semiflows on a complete metric space $(X,d_X)$. Approximations are constructed in terms of reproducing kernel bases that are centered at samples taken along the system trajectory. It is proven that when the samples are dense in a certain type of smooth manifold $M\subseteq X$, the derived rates of convergence depend on the fill distance of samples along the trajectory in that manifold. Error bounds for projection-based and data-dependent approximations of the Koopman operator are derived in the paper. A discussion of how these bounds are realized in intrinsic and extrinsic approximation methods is given. Finally, a numerical example that illustrates qualitatively the convergence guarantees derived in the paper is given.