标题： 关于在再生核希尔伯特空间上Koopman算子的收敛方法 显示英文标题

标题： Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces

摘要： 基于数据的Koopman算子谱分析是对理解众多现实世界动力系统的强大工具，从神经活动到海表温度的变化。 Koopman算子作用于函数空间，并且通常在平方可积函数空间上研究。 然而，在合适的再生核Hilbert空间（RKHS）上定义它具有许多实际优势，包括带误差界的逐点预测、改善的谱性质以促进计算以及更高效的算法，特别是在高维情况下。 我们介绍了首个一般性、可证明收敛的数据驱动算法，用于在RKHS上计算Koopman算子和Perron–Frobenius算子的谱性质。 这些方法高效地计算谱与伪谱，并通过误差控制和谱测度，同时利用RKHS结构来避免在$L^2$设置下所需的大数据分析限制。 函数空间由用户指定的核确定，消除了像$L^2$中基于求积的采样需求，从而在有限的外部提供数据集上实现更大的灵活性。 利用可解复杂性指数层次结构，我们构建了针对这些问题的对抗性动力系统，以表明没有算法能够在更少的限制下成功，从而证明了我们算法的最优性。 值得注意的是，这种不可能性扩展到随机算法和数据集。 我们在来自真实测量和高保真数值模拟的具有挑战性的高维数据集上展示了我们算法的有效性，包括湍流通道流动、结合蛋白的分子动力学、南极海冰浓度以及北半球海面高度。 该算法可在软件包$\texttt{SpecRKHS}$中公开获取。 显示英文摘要

摘要： Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface temperature. The Koopman operator acts on a function space and is most commonly studied on the space of square-integrable functions. However, defining it on a suitable reproducing kernel Hilbert space (RKHS) offers numerous practical advantages, including pointwise predictions with error bounds, improved spectral properties that facilitate computations, and more efficient algorithms, particularly in high dimensions. We introduce the first general, provably convergent, data-driven algorithms for computing spectral properties of Koopman and Perron--Frobenius operators on RKHSs. These methods efficiently compute spectra and pseudospectra with error control and spectral measures while exploiting the RKHS structure to avoid the large-data limits required in the $L^2$ settings. The function space is determined by a user-specified kernel, eliminating the need for quadrature-based sampling as in $L^2$ and enabling greater flexibility with finite, externally provided datasets. Using the Solvability Complexity Index hierarchy, we construct adversarial dynamical systems for these problems to show that no algorithm can succeed in fewer limits, thereby proving the optimality of our algorithms. Notably, this impossibility extends to randomized algorithms and datasets. We demonstrate the effectiveness of our algorithms on challenging, high-dimensional datasets arising from real-world measurements and high-fidelity numerical simulations, including turbulent channel flow, molecular dynamics of a binding protein, Antarctic sea ice concentration, and Northern Hemisphere sea surface height. The algorithms are publicly available in the software package $\texttt{SpecRKHS}$.