标题： 基于算子值RKHS和Koopman逼近的时空预测 显示英文标题

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

摘要： 我们开发了一个全面的框架，用于使用算子值再生核希尔伯特空间（OV RKHS）对时变向量场进行时空预测。 通过将Sobolev正则性与Koopman算子理论相结合，我们建立了基于核的学习动力系统的一致表示定理、逼近速率和谱收敛结果。 我们的理论贡献包括针对时间对齐OV RKHS插值的新表示定理，针对光滑向量场的Sobolev逼近界，核Koopman算子逼近以及谱收敛保证。 这些结果支撑了复杂非线性动力系统的数据驱动降阶建模和预测。 显示英文摘要

摘要： 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.