[论文解读] Modern Koopman Theory for Dynamical Systems

Steven L. Brunton, Marko Budišić+2 more|arXiv (Cornell University)|Feb 24, 2021

Model Reduction and Neural Networks被引用 34

一句话总结

对现代 Koopman 运算符理论在动力系统中的综述，概述通过无限维算子对非线性动力学的线性化、实际算法（特别是 DMD）、理论和数据驱动应用。

ABSTRACT

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.

研究动机与目标

• 解释 Koopman 理论如何通过特征函数与嵌入将非线性动力学线性化。
• 总结运算符理论框架（Koopman、Liouville/Perron–Frobenius）及其与经典动力学系统的联系。
• 描述通过 Dynamic Mode Decomposition (DMD) 和相关算法的实际计算。
• 讨论在大数据和机器学习背景下的应用、挑战与机遇。

提出的方法

• 定义 Koopman 运算符及其连续时间生成器（L）及它们在观测量上的作用。
• 解释 Koopman 运算符的特征函数及它们在线性化动力学中的作用。
• 提出动态模态分解（DMD）作为近似 Koopman 谱的实用数值方法。
• 讨论在更高维空间中产生近似线性动力学的嵌入与坐标变换。
• 将 Koopman 理论与 Perron–Frobenius、Liouville 及 Carleman 线性化框架联系起来。
• 概述 Koopman 理论与数据驱动建模及机器学习技术之间的联系。

实验结果

研究问题

• RQ1 如何通过 Koopman 运算符的特征函数与嵌入将非线性动力学线性表示？
• RQ2 从数据近似 Koopman 谱的理论与实践含义是什么？
• RQ3 DMD 如何与 Koopman 模态分解相关，它的局限性与扩展是什么？
• RQ4 在跨学科的数据驱动 Koopman 分析中的应用、挑战与未来方向是什么？
• RQ5 Koopman 概念如何与控制理论、不确定性量化和传统动力系统理论相连接？

主要发现

• Koopman 理论通过组成算子在无限维的函数空间中对非线性动力学提供线性表示。
• Koopman 运算符的特征函数为分析和预测动力学提供线性化坐标系统。
• DMD 提供一个简单而强大的数值算法，可从数据近似 Koopman 运算符并提取一致结构。
• 该框架与经典动力系统概念（Hartman–Grobman、遍历性）以及概率视角（Perron–Frobenius）相连接。
• 应用遍及流体、流行病学、神经科学、等离子体、金融、机器人和电网，数据驱动方法使实际预测和控制成为可能。