Approximations of the Reproducing Kernel Hilbert Space (RKHS) Embedding Method over Manifolds

Abstract

The reproducing kernel Hilbert space (RKHS) embedding method is a recently introduced estimation approach that seeks to identify the unknown or uncertain function in the governing equations of a nonlinear set of ordinary differential equations (ODEs). While the original state estimate evolves in Euclidean space, the function estimate is constructed in an infinite-dimensional RKHS that must be approximated in practice. When a finite-dimensional approximation is constructed using a basis defined in terms of shifted kernel functions centered at the observations along a trajectory, the RKHS embedding method can be understood as a data-driven approach. This paper derives sufficient conditions that ensure that approximations of the unknown function converge in a Sobolev norm over a submanifold that supports the dynamics. Moreover, the rate of convergence for the finite-dimensional approximations is derived in terms of the fill distance of the samples in the embedded manifold. Numerical simulation of an example problem is carried out to illustrate the qualitative nature of convergence results derived in the paper.

References in corpus (8)

• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator
• An artificial neural network framework for reduced order modeling of transient flows
• Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
• Kernel Approximation on Manifolds II: The $L_{\infty }$-norm of the $L_2$-projector
• Koopman Theory and Linear Approximation Spaces
• Partial Persistence of Excitation in RKHS Embedded Adaptive Estimation
• Koopman Operator Family Spectrum for Nonautonomous Systems - Part 1