# PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator

arXiv: https://arxiv.org/abs/2306.12962 · HTML: https://mlsanity.com/paper/2306.12962/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 88 references that are in the map.
- First posted on arXiv 22 Jun 2023.

## Builds on (most cited first, 30 of 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (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)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction](https://mlsanity.com/paper/2004.00574/) (2020)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019)
- [Koopman operator-based model reduction for switched-system control of PDEs](https://mlsanity.com/paper/1710.06759/) (2017)
- [Koopman Q-learning: Offline Reinforcement Learning via Symmetries of Dynamics](https://mlsanity.com/paper/2111.01365/) (2021)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics](https://mlsanity.com/paper/2010.09961/) (2020)
- [Data-Driven Model Predictive Control using Interpolated Koopman Generators](https://mlsanity.com/paper/2003.07094/) (2020)
- [Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems](https://mlsanity.com/paper/2010.05778/) (2020)

## Built on by (most cited first, 10 of 10)

- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [The Seismic Wavefield Common Task Framework](https://mlsanity.com/paper/2512.19927/) (2025)
- [Universal Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2508.11990/) (2025)
- [CTF4Nuclear: Common Task Framework for Nuclear Fission and Fusion Models](https://mlsanity.com/paper/2605.15549/) (2026)
- [Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations](https://mlsanity.com/paper/2410.10174/) (2024)
- [Information Shapes Koopman Representation](https://mlsanity.com/paper/2510.13025/) (2025)
- [kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learning](https://mlsanity.com/paper/2512.21409/) (2025)
- [Scalable Data-Driven Reachability Analysis and Control via Koopman Operators with Conformal Coverage Guarantees](https://mlsanity.com/paper/2601.01076/) (2026)
- [LC-SAC: Lyapunov-Constrained Soft Actor-Critic via Koopman Operator Theory for Trajectory Tracking and Stabilization](https://mlsanity.com/paper/2602.04132/) (2026)
- [Physics-informed reduced-order modelling with equivariant spectral submanifolds](https://mlsanity.com/paper/2608.04239/) (2026)

## Abstract (arXiv)

PyKoopman is a Python package for the data-driven approximation of the Koopman operator associated with a dynamical system. The Koopman operator is a principled linear embedding of nonlinear dynamics and facilitates the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. In particular, PyKoopman provides tools for data-driven system identification for unforced and actuated systems that build on the equation-free dynamic mode decomposition (DMD) and its variants. In this work, we provide a brief description of the mathematical underpinnings of the Koopman operator, an overview and demonstration of the features implemented in PyKoopman (with code examples), practical advice for users, and a list of potential extensions to PyKoopman. Software is available at http://github.com/dynamicslab/pykoopman