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

• URL: http://arxiv.org/abs/2508.18307v1
• Date: Sat, 23 Aug 2025 04:28:12 GMT
• Title: Learning Spatio-Temporal Dynamics via Operator-Valued RKHS and Kernel Koopman Methods
• Authors: Mahishanka Withanachchi,
• Abstract summary: We introduce a unified framework for learning the evolving-parametric dynamics of valued vector functions by combining operator valued kernel spaces (OV-RKHS) with kernel based Koopman operator methods.<n>The approach enables data driven estimation of complex time vector fields while preserving both spatial and temporal structure.<n>We establish representer theorems for time dependent OV-RKHS, derive Sobolev type approximation for smooth smooth vector bounds, and provide spectral convergence guarantees for kernel Koopman operator approximations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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