Global information about dynamical systems can be extracted by analysing associated infinite-dimensional transfer operators, such as Perron-Frobenius and Koopman operators as well as their infinitesimal generators. In practice, these operators typically need to be approximated from data. Popular approximation methods are extended dynamic mode decomposition (EDMD) and generator extended mode decomposition (gEDMD). We propose a unified framework that leverages Monte Carlo sampling to approximate the operator of interest on a finite-dimensional space spanned by a set of basis functions. Our framework contains EDMD and gEDMD as special cases, but can also be used to approximate more general operators. Our key contributions are proofs of the convergence of the approximating operator and its spectrum under non-restrictive conditions. Moreover, we derive explicit convergence rates and account for the presence of noise in the observations. Whilst all these results are broadly applicable, they also refine previous analyses of EDMD and gEDMD. We verify the analytical results with the aid of several numerical experiments.

翻译：关于动力系统的全局信息可通过分析相关的无穷维传输算子（如Perron-Frobenius算子、Koopman算子及其无穷小生成子）提取。实践中，这些算子通常需要从数据中逼近。常用的逼近方法包括扩展动态模态分解（EDMD）和生成子扩展模态分解（gEDMD）。我们提出一个统一框架，利用蒙特卡洛采样在一组基函数张成的有限维空间上逼近目标算子。该框架以EDMD和gEDMD为特例，同时可逼近更广义的算子。核心贡献在于：在非限制性条件下证明了逼近算子及其谱的收敛性，导出了显式收敛速率，并考虑了观测噪声的影响。这些结果具有广泛适用性，同时改进了先前对EDMD和gEDMD的分析。通过数值实验验证了理论分析结论。