A concise introduction to Koopman operator theory and the Extended Dynamic Mode Decomposition

TL;DR

This paper provides a concise, notation-consistent overview of Koopman operator theory and its algorithmic intersections with Dynamic Mode Decomposition (DMD), Extended Dynamic Mode Decomposition (EDMD), and kernel EDMD. It clarifies how nonlinear dynamics admit a linear representation on observable functions, and how eigenfunctions, eigenvalues, and Koopman modes underpin both theory and practical data-driven approximations. The discussion covers autonomous, non-autonomous, and partially observable systems, introducing delay-embedding (Takens) to handle partial observability and detailing dictionary-based EDMD alongside kernel-based extensions for scalability. By connecting core theory to concrete algorithms, the work aims to provide a clear, usable framework for applying Koopman methods to complex nonlinear systems with consistent notation. The practical impact lies in offering researchers a unified reference that links spectral theory to computable approximations across a range of system classes.

Abstract

The framework of Koopman operator theory is discussed along with its connections to Dynamic Mode Decomposition (DMD) and (Kernel) Extended Dynamic Mode Decomposition (EDMD). This paper provides a succinct overview with consistent notation. The authors hope to provide an exposition that more naturally emphasizes the connections between theory and algorithms which may result in a sense of clarity on the subject.