pykoop

pykoop is a Koopman operator identification library written in Python. It allows the user to specify Koopman lifting functions and regressors in order to learn a linear model of a given system in the lifted space.

To learn more about Koopman operator theory, check out this talk or this review article.

pykoop places heavy emphasis on modular lifting function construction and scikit-learn compatibility. The library aims to make it easy to automatically find good lifting functions and regressor hyperparameters by leveraging scikit-learn's existing cross-validation infrastructure. pykoop also gracefully handles control inputs and multi-episode datasets at every stage of the pipeline.

pykoop also includes several experimental regressors that use linear matrix inequalities to constraint the asymptotic stability of the Koopman system, or regularize the regression using its H-infinity norm. Check out arXiv:2110.09658 [eess.SY] and arXiv:2102.03613 [eess.SY] for details.

Example

Consider Tikhonov-regularized EDMD with polynomial lifting functions applied to mass-spring-damper data. Using pykoop, this can be implemented as:

import pykoop
from sklearn.preprocessing import MaxAbsScaler, StandardScaler

# Get example mass-spring-damper data
eg = pykoop.example_data_msd()

# Create pipeline
kp = pykoop.KoopmanPipeline(
    lifting_functions=[
        ('ma', pykoop.SkLearnLiftingFn(MaxAbsScaler())),
        ('pl', pykoop.PolynomialLiftingFn(order=2)),
        ('ss', pykoop.SkLearnLiftingFn(StandardScaler())),
    ],
    regressor=pykoop.Edmd(alpha=1),
)

# Fit the pipeline
kp.fit(
    eg['X_train'],
    n_inputs=eg['n_inputs'],
    episode_feature=eg['episode_feature'],
)

# Predict using the pipeline
X_pred = kp.predict_trajectory(eg['x0_valid'], eg['u_valid'])

# Score using the pipeline
score = kp.score(eg['X_valid'])

# Plot results
kp.plot_predicted_trajectory(eg['X_valid'], plot_input=True)

More examples are available in examples/, in notebooks/, or on binder.

Library layout

Most of the required classes and functions have been imported into the pykoop namespace. The most important object is the pykoop.KoopmanPipeline, which requires a list of lifting functions and a regressor.

Some example lifting functions are

• pykoop.PolynomialLiftingFn,
• pykoop.RbfLiftingFn,
• pykoop.DelayLiftingFn, and
• pykoop.BilinearInputLiftingFn.

scikit-learn preprocessors can be wrapped into lifting functions using pykoop.SkLearnLiftingFn. States and inputs can be lifted independently using pykoop.SplitPipeline. This is useful to avoid lifting inputs.

Some basic regressors included are

• pykoop.Edmd (includes Tikhonov regularization),
• pykoop.Dmdc, and
• pykoop.Dmd.

More advanced (and experimental) LMI-based regressors are included in the pykoop.lmi_regressors namespace. They allow for different kinds of regularization as well as hard constraints on the Koopman operator.

You can roll your own lifting functions and regressors by inheriting from pykoop.KoopmanLiftingFn, pykoop.EpisodeIndependentLiftingFn, pykoop.EpisodeDependentLiftingFn, and pykoop.KoopmanRegressor.

Some sample dynamic models are also included in the pykoop.dynamic_models namespace.

Installation and testing

pykoop can be installed from PyPI using

$ pip install pykoop

Additional LMI solvers can be installed using

$ pip install mosek
$ pip install cvxopt
$ pip install smcp

Mosek is recommended, but is nonfree and requires a license.

The library can be tested using

$ pip install -r requirements-dev.txt
$ pytest

Note that pytest must be run from the repository's root directory.

To skip unit tests that require a MOSEK license, including all doctests and examples, run

$ pytest ./tests -k "not mosek"

The documentation can be compiled using

$ cd doc
$ make html

If you want a hook to check source code formatting before allowing a commit, you can use

$ cd .git/hooks/
$ ln -s ../../.githooks/pre-commit .
$ chmod +x ./pre-commit

You will need yapf installed for this.

Related packages

Other excellent Python packages for learning dynamical systems exist, summarized in the table below:

Library	Unique features
pykoop	Modular lifting functions Full scikit-learn compatibility Built-in regularization Multi-episode datasets
pykoopman	Continuous-time Koopman operator identification Built-in numerical differentiation Detailed DMD outputs DMDc with known control matrix
PyDMD	Extensive library containing pretty much every variant of DMD
PySINDy	Python implementation of the famous SINDy method Related to, but not the same as, Koopman operator approximation

Citation

If you use this software in your research, please cite it as below or see CITATION.cff.

@software{dahdah_pykoop_2024,
    title={{decargroup/pykoop}},
    doi={10.5281/zenodo.5576490},
    url={https://github.com/decargroup/pykoop},
    publisher={Zenodo},
    author={Steven Dahdah and James Richard Forbes},
    version = {{v2.0.2}},
    year={2024},
}

License

This project is distributed under the MIT License, except the contents of pykoop/_sklearn_metaestimators/, which are from the scikit-learn project, and are distributed under the BSD-3-Clause License.