论文标题
用于数据驱动计算的动力学系统计算的MPEDMD算法
The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems
论文作者
论文摘要
Koopman运算符全球线性化非线性动力学系统及其光谱信息是非线性动力学系统分析和分解的强大工具。但是，Koopman操作员是无限维度的，计算其光谱信息是一个巨大的挑战。我们介绍了Measure-teardended动态模式分解（$ \ texttt {mpedmd} $），这是第一种截断方法，其特征性组件收敛到koopman运算符的光谱量，用于一般测量的动态系统。 $ \ texttt {mpedmd} $是基于正交式procrustes问题的数据驱动算法，该问题使用可观察的一般字典来强制测量Koopman运算符的截断。它具有灵活性且易于使用的任何预先存在的DMD类型方法，并且具有不同类型的数据。对于投影值和标量值的光谱措施，光谱和Koopman模式分解，我们证明了$ \ texttt {mpedmd} $的融合。对于延迟嵌入（Krylov子空间）的情况，我们的结果包括随着字典的大小增加，光谱测量近似值的第一个收敛速率。 We demonstrate $\texttt{mpEDMD}$ on a range of challenging examples, its increased robustness to noise compared with other DMD-type methods, and its ability to capture the energy conservation and cascade of experimental measurements of a turbulent boundary layer flow with Reynolds number $> 6\times 10^4$ and state-space dimension $>10^5$.
Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite-dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition ($\texttt{mpEDMD}$), the first truncation method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. $\texttt{mpEDMD}$ is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any pre-existing DMD-type method, and with different types of data. We prove convergence of $\texttt{mpEDMD}$ for projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include the first convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate $\texttt{mpEDMD}$ on a range of challenging examples, its increased robustness to noise compared with other DMD-type methods, and its ability to capture the energy conservation and cascade of experimental measurements of a turbulent boundary layer flow with Reynolds number $> 6\times 10^4$ and state-space dimension $>10^5$.
