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

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.