Spatio-Temporal Prediction via Operator-Valued RKHS and Koopman Approximation

TL;DR

This paper introduces a new framework combining operator-valued RKHS and Koopman theory for accurate spatio-temporal prediction of complex dynamical systems, with rigorous theoretical guarantees.

Contribution

It develops novel representer theorems, approximation bounds, and spectral convergence results for kernel-based learning of dynamical systems using OV RKHS and Koopman operators.

Findings

Established new representer theorems for OV RKHS interpolation.

Derived Sobolev approximation bounds for smooth vector fields.

Proved spectral convergence guarantees for kernel Koopman operator approximations.

Abstract

We develop a comprehensive framework for spatio-temporal prediction of time-varying vector fields using operator-valued reproducing kernel Hilbert spaces (OV RKHS). By integrating Sobolev regularity with Koopman operator theory, we establish representer theorems, approximation rates, and spectral convergence results for kernel-based learning of dynamical systems. Our theoretical contributions include new representer theorems for time-aligned OV RKHS interpolation, Sobolev approximation bounds for smooth vector fields, kernel Koopman operator approximations, and spectral convergence guarantees. These results underpin data-driven reduced-order modeling and forecasting for complex nonlinear dynamical systems.