This paper examines the application of the Kernel Sum of Squares (KSOS) method for enhancing kernel learning from data, particularly in the context of dynamical systems. Traditional kernel-based methods, despite their theoretical soundness and numerical efficiency, frequently struggle with selecting optimal base kernels and parameter tuning, especially with gradient-based methods prone to local optima. KSOS mitigates these issues by leveraging a global optimization framework with kernel-based surrogate functions, thereby achieving more reliable and precise learning of dynamical systems. Through comprehensive numerical experiments on the Logistic Map, Henon Map, and Lorentz System, KSOS is shown to consistently outperform gradient descent in minimizing the relative-$\rho$ metric and improving kernel accuracy. These results highlight KSOS's effectiveness in predicting the behavior of chaotic dynamical systems, demonstrating its capability to adapt kernels to underlying dynamics and enhance the robustness and predictive power of kernel-based approaches, making it a valuable asset for time series analysis in various scientific fields.

翻译：本文研究了核平方和（KSOS）方法在增强从数据中进行核学习方面的应用，特别是在动力系统背景下。传统的基于核的方法尽管具有理论上的严谨性和数值效率，但在选择最优基核和参数调优方面常常面临困难，尤其是基于梯度的方法容易陷入局部最优解。KSOS通过利用基于核的代理函数构建全局优化框架来缓解这些问题，从而实现对动力系统更可靠、更精确的学习。通过对Logistic映射、Henon映射和Lorenz系统进行的全面数值实验表明，KSOS在最小化相对-$\rho$度量及提升核精度方面持续优于梯度下降法。这些结果突显了KSOS在预测混沌动力系统行为方面的有效性，展示了其使核适应底层动力学并增强基于核方法的鲁棒性和预测能力，使其成为各科学领域中时间序列分析的宝贵工具。