SIAM Review, Volume 64, Issue 2, Page 227-227, May 2022.
Steven L. Brunton, Marko Budišić, Eurika Kaiser, and J. Nathan Kutz are the authors of the Survey and Review paper in this issue, “Modern Koopman Theory for Dynamical Systems.” Koopman theory is a valuable formalism for the study of dynamical systems; it has gained popularity in recent years in connection with data-driven analysis, control theory, and other areas. The basic idea is simple. If $(d/dt) {x} = {f}({x})$ is a dynamical system in ${\mathbb R}^n$, its solution flow is the one-parameter family of maps ${F}^t: {\mathbb R}^n\rightarrow {\mathbb R}^n$ such that $t\mapsto {F}^t({x}_0)$ is the solution that at time $t=0$ takes the value ${x}_0$. In an alternative description, rather than looking at points in ${\mathbb R}^n$ being moved by the flow, one may consider real-valued functions $g({x})$ being transformed by the dynamical system as $g \mapsto {\cal K}^t(g)$, where the Koopman (or composition) operator ${\cal K}^t$ is defined by ${\cal K}(g)({x}) = g({F}^t({x}))$. Thus the Koopman operator acts on an infinite-dimensional space of functions $g$ (bad news), but it is linear, even if the original dynamical system is not (good news). The idea behind the operator ${\cal K}^t$ is particularly useful when the dynamical system is being studied by big data$/$machine learning techniques in cases where $f$ and ${F}^t$ are unknown but measurements $g_i({x}(t_j))$ along a solution ${x}(t)$ are available. While the use of composition operators is very much older, B. O. Koopman noted in a seminal 1931 paper that, in the particular case of conservative dynamics, the operator ${\cal K}^t$ will be unitary in a suitable $L^2$ space, an observation that allowed him to apply to classical mechanics the theory of Hilbert space operators being developed at the time to formulate mathematically quantum mechanics. Koopman's results have been extended in many directions to dissipative or conservative systems, in continuous or discrete time. The survey in this issue does not assume a previous knowledge of Koopman theory and reviews recent developments on Koopman operators, with particular emphasis on the dynamic mode decomposition (DMD) algorithm and its variants used in many real-life applications. The paper contains almost five hundred references, shows applications to fluid mechanics, epidemiology, neuroscience, plasma physics, finance, robotics, the power grid, and other fields, and discusses connections with many areas, including numerical linear algebra, control theory, statistics, model reduction, and uncertainty qualification. I believe it will be of interest to a wide range of readers.

中文翻译：

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调查和审查

SIAM 评论，第 64 卷，第 2 期，第 227-227 页，2022 年 5 月。
Steven L. Brunton、Marko Budišić、Eurika Kaiser 和 J. Nathan Kutz 是本期“现代 Koopman 动力系统理论”的调查和评论论文的作者。考夫曼理论是动力系统研究的一种有价值的形式主义；近年来，它在数据驱动分析、控制理论和其他领域得到普及。基本思想很简单。如果 $(d/dt) {x} = {f}({x})$ 是 ${\mathbb R}^n$ 中的一个动力系统，它的解流是单参数映射族 ${F} ^t: {\mathbb R}^n\rightarrow {\mathbb R}^n$ 使得 $t\mapsto {F}^t({x}_0)$ 是在 $t=0$ 时的解价值 ${x}_0$。在另一种描述中，与其查看 ${\mathbb R}^n$ 中被流移动的点，可以将动态系统将实值函数 $g({x})$ 转换为 $g \mapsto {\cal K}^t(g)$，其中 Koopman（或合成）运算符 ${\cal K}^t$ 由 ${\cal K}(g)({x}) = g({F}^t({x}))$ 定义。因此，Koopman 算子作用于函数 $g$ 的无限维空间（坏消息），但它是线性的，即使原始动力系统不是（好消息）。当 $f$ 和 ${F}^t$ 未知但沿着解 ${x}(t)$ 的测量 $g_i({x}(t_j))$ 是可用的。虽然组合运算符的使用非常古老，但 BO Koopman 在 1931 年的一篇开创性论文中指出，在保守动力学的特定情况下，算子 ${\cal K}^t$ 在合适的 $L^2$ 空间中是酉的，这一观察使他能够将当时正在发展的希尔伯特空间算子理论应用于经典力学，以制定数学量子力学. Koopman 的结果已在许多方向上扩展到耗散或保守系统，在连续或离散时间内。本期的调查不假设以前了解 Koopman 理论，而是回顾 Koopman 算子的最新发展，特别强调动态模式分解 (DMD) 算法及其在许多实际应用中使用的变体。该论文包含近五百篇参考文献，展示了在流体力学、流行病学、神经科学、等离子体物理学、金融、机器人技术、电网等领域的应用，并讨论了与许多领域的联系，包括数值线性代数、控制理论、统计学、模型简化和不确定性限定。我相信它会引起广大读者的兴趣。