Modern Koopman Theory for Dynamical Systems

The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing data science. First-principles derivations 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 past decade, which nonlinear dynamics represented terms an infinite-dimensional linear operator acting on space all possible measurement functions system. This representation tremendous potential enable prediction, estimation, control with standard textbook methods developed for systems. However, obtaining finite-dimensional coordinate embeddings appear approximately remains central open challenge. success analysis due primarily three key factors: (1) there exists rigorous connecting it classical geometric systems; (2) approach formulated measurements, making ideal leveraging big machine learning techniques; (3) simple, yet powerful numerical algorithms, such dynamic mode decomposition (DMD), have been extended reduce practice real-world applications. In this review, we provide overview theory, describing recent theoretical algorithmic developments highlighting these diverse range We also discuss advances challenges rapidly growing likely drive future significantly transform landscape