Modern Koopman Theory for Dynamical Systems

TL;DR

This review surveys modern Koopman operator theory for dynamical systems, emphasizing data-driven embeddings and finite-dimensional approximations via DMD and its extensions. It connects operator-theoretic foundations (Koopman and Perron–Frobenius) with practical computational tools (DMD, EDMD, HAVOK, diffusion maps) and shows how lifted observables enable linear treatment of nonlinear dynamics. The article covers nonautonomous and stochastic extensions, PDE connections, and a broad array of control strategies (DMDc, eDMDc, bilinear and intrinsic Koopman controllers) with applications across fluids, neuroscience, epidemiology, and engineering. It also discusses convergence, identifiability of Koopman embeddings, and the future role of neural networks and tensor methods in scalable, robust data-driven control and analysis of complex systems.

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.

Figures (27)

• Figure 1: Different Koopman perspectives for the Duffing oscillator, $\ddot{x} = x-x^3$, the equation for a particle in a double potential well. (a) Traditional linearization near the fixed points gives small regions where the system is approximately linear. (b) Koopman theory may extend the Hartman--Grobman theorem to enlarge the domain of linearity until the next fixed point lan2013physd. (c) There are also global Koopman eigenfunctions, like the Hamiltonian energy, although these lose information about which basin the solution is in. (d) Yet a third perspective seeks a coordinate transformation to rescale space and time until dynamics live on a hypertoroid.
• Figure 1: Koopman eigenfunctions constructed as \ref{['eq:eigenvectors-to-eigenfunctions']} for a linear matrix ODE $\dot\mathbf{x} = \mathbf{A} \mathbf{x}$ with a saddle-type fixed point at the origin. Velocity field and sample orbits are overlaid in white.
• Figure 1: Overview of DMD illustrated on the fluid flow past a circular cylinder at Reynolds number $100$. Reproduced from Kutz et al. kutz2016book.
• Figure 1: Isostables and isochrons for the FitzHugh--Nagumo model acting as a deformed rectifiable coordinate system in vicinity of a focus (left) and node (right). Reproduced with permission, from Mauroy et al. 2013 Physica D mauroy2013.
• Figure 1: Decomposition of chaos into a linear system with forcing. A time series $x(t)$ is stacked into a Hankel matrix $\mathbf{H}$. The SVD of $\mathbf{H}$ yields a hierarchy of eigen time series that produce a delay-embedded attractor. A best-fit linear regression model is obtained on the delay coordinates $\mathbf{v}$; the linear fit for the first $r-1$ variables is excellent, but the last coordinate $v_r$ is not well-modeled as linear. Instead, $v_r$ is an input that forces the first ${r-1}$ variables. Rare forcing events correspond to lobe switching in the chaotic dynamics. This architecture is called the Hankel alternative view of Koopman (HAVOK) analysis, from brunton2017natcomm. Figure modified from Brunton et al. brunton2017natcomm.

Theorems & Definitions (3)

• Theorem 1
• Proposition 2
• Theorem 1