PyKoopman

PyKoopman is a Python package for computing data-driven approximations to the Koopman operator.

Data-driven approximation of Koopman operator

Given a nonlinear dynamical system,

x'(t) = f(x(t)),

the Koopman operator governs the temporal evolution of the measurement function. Unfortunately, it is an infinite-dimensional linear operator. Most of the time, one has to project the Koopman operator onto a finite-dimensional subspace that is spanned by user-defined/data-adaptive functions.

z = \Phi(x).

If the system state is also contained in such subspace, then effectively, the nonlinear dynamical system is (approximately) linearized in a global sense.

The goal of data-driven approximation of Koopman operator is to find such a set of functions that span such lifted space and the transition matrix associated with the lifted system.

Structure of PyKoopman

PyKoopman package is centered around the Koopman class and KoopmanContinuous class. It consists of two key components

• observables: a set of observables functions, which spans the subspace for projection.
• regressor: the optimization algorithm to find the best fit for the projection of Koopman operator.

After Koopman/KoopmanContinuous object has been created, it must be fit to data, similar to a scikit-learn model. We design PyKoopman such that it is compatible to scikit-learn objects and methods as much as possible.

Features implemented

• Observable library for lifting the state into the observable space
  • Identity (for DMD/DMDc or in case users want to compute observables themselves): Identity
  • Multivariate polynomials: Polynomial
  • Time delay coordinates: TimeDelay
  • Radial basis functions: RadialBasisFunctions
  • Random Fourier features: RandomFourierFeatures
  • Custom library (defined by user-supplied functions): CustomObservables
  • Concatenation of observables: ConcatObservables
• System identification method for performing regression
  • Dynamic mode decomposition: PyDMDRegressor
  • Dynamic mode decomposition with control: DMDc
  • Extended dynamic mode decomposition: EDMD
  • Extended dynamic mode decomposition with control: EDMDc
  • Kernel dynamic mode decomposition: KDMD
  • Hankel Alternative View of Koopman Analysis: HAVOK
  • Neural Network DMD: NNDMD
• Sparse construction of Koopman invariant subspace
  • Multi-task learning based on linearity consistency

Examples

1. Learning how to create observables

2. Learning how to compute time derivatives

3. Dynamic mode decomposition on two mixed spatial signals

4. Dynamic mode decomposition with control on a 2D linear system

5. Dynamic mode decomposition with control (DMDc) for a 128D system

6. Dynamic mode decomposition with control on a high-dimensional linear system

7. Successful examples of using Dynamic mode decomposition on PDE system

8. Unsuccessful examples of using Dynamic mode decomposition on PDE system

9. Extended DMD for Van Der Pol System

10. Learning Koopman eigenfunctions on Slow manifold

11. Comparing DMD and KDMD for Slow manifold dynamics

12. Extended DMD with control for chaotic duffing oscillator

13. Extended DMD with control for Van der Pol oscillator

14. Hankel Alternative View of Koopman Operator for Lorenz System

15. Hankel DMD with control for Van der Pol Oscillator

16. Neural Network DMD on Slow Manifold

17. EDMD and NNDMD for a simple linear system

18. Sparisfying a minimal Koopman invariant subspace from EDMD for a simple linear system

Installation

Installing with pip

If you are using Linux or macOS you can install PyKoopman with pip:

pip install pykoopman

Installing from source

First clone this repository:

git clone https://github.com/dynamicslab/pykoopman

Second, it is highly recommended to use venv to get a local python environment

python -m venv venv
source ./venv/bin/activate

In windows, you activate virtual environment in a different way

.\venv\Scripts\activate

Then, to install the package, run

python -m pip install -e .

If you do not have root access, you should add the --user option to the above lines.

Documentation

The documentation for PyKoopman is hosted on Read the Docs.

Community guidelines

Contributing code

We welcome contributions to PyKoopman. To contribute a new feature please submit a pull request. To get started we recommend installing the packages in "developer mode" via

python -m pip install -e .[dev]

This will allow you to run unit tests and automatically format your code. To be accepted your code should conform to PEP8 and pass all unit tests. Code can be tested by invoking

pytest

We recommed using pre-commit to format your code. Once you have staged changes to commit

git add path/to/changed/file.py

you can run the following to automatically reformat your staged code

pre-commit -a -v

Note that you will then need to re-stage any changes pre-commit made to your code.

Reporting issues or bugs

If you find a bug in the code or want to request a new feature, please open an issue.

Citing PyKoopman

@misc{pan2023pykoopman,
      title={PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator},
      author={Shaowu Pan and Eurika Kaiser and Brian M. de Silva and J. Nathan Kutz and Steven L. Brunton},
      year={2023},
      eprint={2306.12962},
      archivePrefix={arXiv},
      primaryClass={eess.SY}
}

Related packages

• PySINDy - A Python libray for the Sparse Identification of Nonlinear Dynamical systems (SINDy) method introduced in Brunton et al. (2016a).
• Deeptime - A Python library for the analysis of time series data with methods for dimension reduction, clustering, and Markov model estimation.
• PyDMD - A Python package using the Dynamic Mode Decomposition (DMD) for a data-driven model simplification based on spatiotemporal coherent structures. DMD is a great alternative to SINDy.
• pykoop - a Koopman operator identification library written in Python
• DLKoopman - a deep learning library for Koopman operator