Modern Koopman theory for dynamical systems

SL Brunton, M Budišić, E Kaiser, JN Kutz - arXiv preprint arXiv:2102.12086, 2021 - arxiv.org
The field of dynamical systems is being transformed by the mathematical tools and
algorithms emerging from modern computing and data science. First-principles derivations …

Koopman operators for estimation and control of dynamical systems

SE Otto, CW Rowley - Annual Review of Control, Robotics, and …, 2021 - annualreviews.org
A common way to represent a system's dynamics is to specify how the state evolves in time.
An alternative viewpoint is to specify how functions of the state evolve in time. This evolution …

Residual dynamic mode decomposition: robust and verified Koopmanism

MJ Colbrook, LJ Ayton, M Szőke - Journal of Fluid Mechanics, 2023 - cambridge.org
Dynamic mode decomposition (DMD) describes complex dynamic processes through a
hierarchy of simpler coherent features. DMD is regularly used to understand the …

The mpEDMD algorithm for data-driven computations of measure-preserving dynamical systems

MJ Colbrook - SIAM Journal on Numerical Analysis, 2023 - SIAM
Koopman operators globally linearize nonlinear dynamical systems and their spectral
information is a powerful tool for the analysis and decomposition of nonlinear dynamical …

Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems

MJ Colbrook, A Townsend - Communications on Pure and …, 2024 - Wiley Online Library
Koopman operators are infinite‐dimensional operators that globally linearize nonlinear
dynamical systems, making their spectral information valuable for understanding dynamics …

Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems

MJ Colbrook, Q Li, RV Raut, A Townsend - Nonlinear Dynamics, 2024 - Springer
Koopman operators linearize nonlinear dynamical systems, making their spectral
information of crucial interest. Numerous algorithms have been developed to approximate …

[HTML][HTML] Error bounds for kernel-based approximations of the Koopman operator

FM Philipp, M Schaller, K Worthmann, S Peitz… - Applied and …, 2024 - Elsevier
We consider the data-driven approximation of the Koopman operator for stochastic
differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the …

Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques

R Alexander, D Giannakis - Physica D: Nonlinear Phenomena, 2020 - Elsevier
Kernel analog forecasting (KAF), alternatively known as kernel principal component
regression, is a kernel method used for nonparametric statistical forecasting of dynamically …

Limits and powers of Koopman learning

MJ Colbrook, I Mezić, A Stepanenko - arXiv preprint arXiv:2407.06312, 2024 - arxiv.org
Dynamical systems provide a comprehensive way to study complex and changing behaviors
across various sciences. Many modern systems are too complicated to analyze directly or …

Spectral analysis of climate dynamics with operator-theoretic approaches

G Froyland, D Giannakis, BR Lintner, M Pike… - Nature …, 2021 - nature.com
The Earth's climate system is a classical example of a multiscale, multiphysics dynamical
system with an extremely large number of active degrees of freedom, exhibiting variability on …