Modern Koopman Theory for Dynamical Systems
Modern Koopman Theory for Dynamical Systems
复制标题
动力系统的现代库普曼理论
DOI：
10.1137/21m1401243
复制
发表时间：
2022-06-01
期刊：
影响因子：
10.2
通讯作者：
Kutz, J. Nathan
中科院分区：
文献类型：
--
作者：
Brunton, Steven L.;Budisic, Marko;Kutz, J. Nathan
现代计算和数据科学中涌现出的数学工具和算法正在改变动力系统领域。第一原理推导和渐近归约正在让位于数据驱动的方法，这些方法在算符理论或概率框架中制定模型。在过去的十年里，库普曼谱理论已经成为一种占主导地位的观点，其中非线性动力学用作用于系统所有可能的测量函数的空间上的无限维线性算子来表示。这种非线性动力学的线性表示具有巨大的潜力，可以用为线性系统开发的标准教科书方法来预测、估计和控制非线性系统。然而，获得动力学看起来近似线性的有限维坐标系和嵌入仍然是一个中心开放挑战。库普曼分析的成功主要归功于三个关键因素：(1)存在将其与动力系统的经典几何方法联系起来的严谨理论；(2)该方法是根据测量制定的，使其成为利用大数据和机器学习技术的理想选择；以及(3)开发和扩展了简单而强大的数值算法，如动态模式分解(DMD)，以将库普曼理论简化为实际应用。在这篇综述中，我们提供了现代库普曼算子理论的概述，描述了最近的理论和算法的发展，并强调了这些方法具有不同的应用范围。我们还讨论了快速增长的机器学习领域的关键进展和挑战，这些进展和挑战可能会推动未来的发展，并显著改变动力系统的理论格局。
The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator-theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: (1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems; (2) the approach is formulated in terms of measurements, making it ideal for leveraging big data and machine learning techniques; and (3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.