# Koopman operator theory: fundamentals, control, and applications

arXiv: https://arxiv.org/abs/2607.01819 · HTML: https://mlsanity.com/paper/2607.01819/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 92 of its 198 references that are in the map.
- First posted on arXiv 2 Jul 2026.

## Builds on (most cited first, 30 of 92)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [A Survey of Deep Learning and Foundation Models for Time Series Forecasting](https://mlsanity.com/paper/2401.13912/) (2024)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Koopman Q-learning: Offline Reinforcement Learning via Symmetries of Dynamics](https://mlsanity.com/paper/2111.01365/) (2021)
- [Finite-data error bounds for Koopman-based prediction and control](https://mlsanity.com/paper/2108.07102/) (2021)
- [The Multiverse of Dynamic Mode Decomposition Algorithms](https://mlsanity.com/paper/2312.00137/) (2023)

## Built on by (most cited first, 1 of 1)

- [Learning to control switching nonlinear systems with Koopman operator regression](https://mlsanity.com/paper/2607.11344/) (2026)

## Abstract (arXiv)

The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.