Learning Spatio-Temporal Dynamics via Operator-Valued RKHS and Kernel Koopman Methods
This provides theoretically grounded tools for forecasting, control, and uncertainty quantification in spatio-temporal machine learning, addressing complex dynamics in domains like fluid mechanics or climate modeling.
The paper tackles the problem of learning spatio-temporal dynamics in high-dimensional nonlinear systems by introducing a unified framework combining operator-valued RKHS and kernel Koopman methods, enabling nonparametric estimation with theoretical guarantees for approximation and spectral convergence.
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.