Convergence Rates for Approximations of Deterministic Koopman Operators via Inverse Problems

This paper derives rates of convergence of approximations of the deterministic Koopman operator using a framework based on estimating solutions of inverse problems. By restricting the domain of the Koopman operator, simple sufficient conditions are derived that ensure that the resulting Koopman operator is compact when acting on a suitable reproducing kernel Hilbert space (RKHS). Approximations of the Koopman operator, or its inverse, are derived in terms of Galerkin approximations of solutions to an associated inverse problem which depends on noisy data. The resulting bounds on accuracy of approximations to the Koopman operator then take a classical form: we obtain explicit representations of the contributions of the approximation error and the generalization error in terms of the reduced dimension and noise level. As the reduced dimension of the approximations increases, the approximation error decreases, while the generalization error increases. The generalization error increases with an increase in the noise level. In the case of a discrete evolution over a smooth, compact, connected, Riemannian manifold, we show that these two contributions to the error can be bounded in terms of the fill distance of centers of approximation and samples in the manifold.