  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.
