Learning Spatio-Temporal Dynamics via Operator-Valued RKHS and Kernel Koopman Methods

TL;DR

This paper presents a unified, theoretically grounded framework combining operator-valued RKHS and kernel Koopman methods for nonparametric learning and prediction of complex spatio-temporal vector fields in high-dimensional systems.

Contribution

It introduces a novel integration of OV-RKHS with kernel Koopman operators, providing new theoretical results and enabling efficient modeling of spatio-temporal dynamics.

Findings

Establishes representer theorems for time-dependent OV-RKHS interpolation.

Derives Sobolev type approximation bounds for smooth vector fields.

Provides spectral convergence guarantees for kernel Koopman operator approximations.

Abstract

We introduce a unified framework for learning the spatio-temporal dynamics of vector valued functions by combining operator valued reproducing kernel Hilbert spaces (OV-RKHS) with kernel based Koopman operator methods. The approach enables nonparametric and data driven estimation of complex time evolving vector fields while preserving both spatial and temporal structure. We establish representer theorems for time dependent OV-RKHS interpolation, derive Sobolev type approximation bounds for smooth vector fields, and provide spectral convergence guarantees for kernel Koopman operator approximations. This framework supports efficient reduced order modeling and long term prediction of high dimensional nonlinear systems, offering theoretically grounded tools for forecasting, control, and uncertainty quantification in spatio-temporal machine learning.