用核方法学习时空动态，能高效预测复杂系统演化。
Learning Spatio-Temporal Dynamics via Operator-Valued RKHS and Kernel Koopman Methods
- 结合算子型核空间与谱方法，非参数建模时空向量场。
- 理论保证逼近误差和谱收敛性，支持长期预测。
- 适合需要高维非线性系统建模的科研人员使用。
我们提出一种统一框架，通过算子值再生核希尔伯特空间（OV-RKHS）与基于核的Koopman算子方法，联合学习向量值函数的时空动力学。该方法实现对复杂时变向量场的非参数化、数据驱动估计，同时保留空间与时间结构。我们建立了时变OV-RKHS插值的表示定理，推导了光滑向量场的Sobolev型逼近界，并提供了核Koopman算子近似的谱收敛保证。该框架支持高效的降维建模与高维非线性系统的长期预测，为时空机器学习中的预报、控制与不确定性量化提供理论坚实工具。
原文摘要 · Abstract (English)
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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