流形上Koopman算子的内、外逼近
Intrinsic and Extrinsic Approximation of Koopman Operators over Manifolds
摘要：
本文在完全度量空间$(X,d_X)$上导出了与离散的、确定的、连续的半流有关的Koopman算子的某些逼近的收敛速度。近似构造的再生核基地，集中在沿系统轨迹的样本。证明了当样本在某种类型的光滑流形$M\subseteq X$中稠密时，导出的收敛速度依赖于样本沿该流形中轨迹的填充距离。本文导出了Koopman算子的基于投影和依赖于数据的近似的误差界。讨论如何实现这些界限的内在和外在的近似方法。最后给出了一个数值例子，定性地说明了文中所导出的收敛保证。
英文摘要：
This paper derives rates of convergence of certain approximations of the Koopman operators that are associated with discrete, deterministic, continuous semiflows on a complete metric space $(X,d_X)$. Approximations are constructed in terms of reproducing kernel bases that are centered at samples taken along the system trajectory. It is proven that when the samples are dense in a certain type of smooth manifold $M\subseteq X$, the derived rates of convergence depend on the fill distance of samples along the trajectory in that manifold. Error bounds for projection-based and data-dependent approximations of the Koopman operator are derived in the paper. A discussion of how these bounds are realized in intrinsic and extrinsic approximation methods is given. Finally, a numerical example that illustrates qualitatively the convergence guarantees derived in the paper is given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
