[Paper Review] Modern Koopman Theory for Dynamical Systems

Steven L. Brunton, Marko Budišić+2 more|arXiv (Cornell University)|Feb 24, 2021

Model Reduction and Neural Networks34 citations

TL;DR

A survey of modern Koopman operator theory for dynamical systems, outlining the linearization of nonlinear dynamics via infinite-dimensional operators, practical algorithms (notably DMD), theory, and data-driven applications.

ABSTRACT

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.

Motivation & Objective

• Explain how Koopman theory linearizes nonlinear dynamics via eigenfunctions and embeddings.
• Summarize the operator-theoretic framework (Koopman, Liouville/Perron–Frobenius) and its connections to classical dynamical systems.
• Describe practical computation via Dynamic Mode Decomposition (DMD) and related algorithms.
• Discuss applications, challenges, and opportunities in big-data and machine learning contexts.

Proposed method

• Define the Koopman operator and its continuous-time generator (L) and their action on observables.
• Explain eigenfunctions of the Koopman operator and their role in linearizing dynamics.
• Present the dynamic mode decomposition (DMD) as a practical numerical method to approximate the Koopman spectrum.
• Discuss embeddings and coordinate transformations that yield approximately linear dynamics in higher-dimensional spaces.
• Relate Koopman theory to Perron–Frobenius, Liouville, and Carleman linearization frameworks.
• Outline connections between Koopman theory and data-driven modeling and machine learning techniques.

Experimental results

Research questions

• RQ1 How can nonlinear dynamics be represented linearly via Koopman operator eigenfunctions and embeddings?
• RQ2 What are the theoretical and practical implications of approximating the Koopman spectrum from data?
• RQ3 How does DMD relate to Koopman mode decomposition and what are its limitations and extensions?
• RQ4 What are the applications, challenges, and future directions for data-driven Koopman analysis across disciplines?
• RQ5 How do Koopman concepts connect to control theory, uncertainty quantification, and traditional dynamical systems theory?

Key findings

• Koopman theory offers a linear representation of nonlinear dynamics on an infinite-dimensional function space via composition operators.
• Eigenfunctions of the Koopman operator provide a linearizing coordinate system for analyzing and predicting dynamics.
• DMD provides a simple, powerful numerical algorithm to approximate the Koopman operator from data and extract coherent structures.
• The framework connects to classical dynamical systems concepts (Hartman–Grobman, ergodicity) and to probabilistic perspectives (Perron–Frobenius).
• Applications span fluids, epidemiology, neuroscience, plasmas, finance, robotics, and power grids, with data-driven methods enabling practical prediction and control.