Survey and Review

Abstract

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.