A method that is employed to evaluate a Koopman matrix from a data set of snapshot pairs is the extended dynamical mode decomposition (EDMD). The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of observables, and is beneficial when employed in the analysis of dynamics. The Koopman matrix corresponds to an approximation of the Koopman operator, requiring a specific dictionary to represent the operator. In this study, an alternative approach for evaluating the Koopman matrix for stochastic differential equations has been proposed. Using the system equations the Koopman matrix can be directly derived without any sampling. Hence, this approach is complementary to a data-driven approach provided a prior knowledge of the system equations is available. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD have also been demonstrated considering a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.

翻译：用于从截图配对数据集中评价Koopman矩阵的方法是扩展动态模式分解(EDMD)的扩展动态模式。Koopman操作员是一个线性但无限的操作员,它控制着可观测到的演变,在进行动态分析时是有用的。Koopman矩阵与Koopman操作员的近似值相对应,需要用特定的字典来代表操作员。在这项研究中,提出了一种用于评价Koopman矩阵的蒸馏式差异方程式的替代方法。使用系统方程式可以直接得出Koopman矩阵,而无需任何抽样。因此,这一方法是对数据驱动法的补充,但必须事先掌握对系统方程式的了解。拟议方法包括组合法、固态近似法和外推法。与EDMD的比较也已经证明要考虑一个温暖的van der Pol系统。拟议的方法产生合理的结果,即使在EDMD显示缓慢的趋同行为的情况下也是如此。