Approximation of discrete and orbital Koopman operators over subsets and manifolds

This paper introduces a kernel-based approach for constructing approximations of the Koopman operators for semiflows in discrete time and orbital Koopman operators for continuous time semiflows. The primary advantage of the proposed construction is that the approximations follow certain rates of convergence which are dependent on how data samples fill certain subsets of the state space. In particular, we derive the rate of convergence for two scenarios: (1) the data samples Ξ are dense in a compact state space X , and (2) the data samples Ξ are dense in a limiting set Ω contained in an Euclidean space. Two general classes of Koopman operator approximations are considered in this paper, referred to as projection-based approximation and data-driven approximation. Projection-based approximations assume that the underlying dynamics governing the discrete or continuous time semiflows is known. On the other hand, data-driven approximations rely samples of the semiflow states to approximate the Koopman operator. In both types of approximations, the regularity of the underlying set and the smoothness of the space of functions on which the Koopman operator acts determine the rates of approximations. In the strongest error bounds derived in the paper, it is shown that the error in approximation of the Koopman operator decays like O ( h Ω n , Ω p ) , where h Ω n , Ω is the fill rate of the samples Ω n in the limiting set Ω and p is an exponent related to the choice of the kernel and the smoothness of functions on which the Koopman operator acts. Such error bounds are obtained when either the limiting subset Ω = X , when it is a proper subset Ω ⊂ X that is sufficiently regular, or when it is a type of smooth manifold Ω = M .