AutoKoopman

Overview

AutoKoopman is a high-level system identification tool that automatically optimizes all hyper-parameters to estimate accurate system models with globally linearized representations. Implemented as a python library under shared class interfaces, AutoKoopman uses a collection of Koopman-based algorithms centered on conventional dynamic mode decomposition and deep learning. Koopman theory relies on embedding system states to observables; AutoKoopman provides major types of static observables.

The library supports * Discrete-Time and Continuous-Time System Identification * Extended Dynamic Mode Decomposition (EDMD) [Williams et al.] * Deep Koopman [Li et al.] * SINDy [Brunton et al.] * Static Observables * Random Fourier Features [Bak et al.] * Polynomial * Neural Network [Li et al.] * System Identification with Input and Control * Koopman with Input and Control (KIC) [Proctor et al.] * Online (Streaming) System Identification * Online DMD [Zhang et al.] * Hyperparameter Optimization * Random Search * Grid Search * Bayesian Optimization

Use Cases

The library is intended for a systems engineer / researcher who wishes to leverage data-driven dynamical systems techniques. The user may have measurements of their system with no prior model.

• Prediction: Predict the evolution of a system over long time horizons
• Control: Synthesize control signals that achieve desired closed-loop behaviors and are optimal with respect to some objective.
• Verification: Prove or falsify the safety requirements of a system.

Installation

The module is published on PyPI. It requires python 3.8 or higher. With pip installed, run shell pip install autokoopman at the repo root. Run shell python -c "import autokoopman" to ensure that the module can be imported.

Examples

A Complete Example

AutoKoopman has a convenience function auto_koopman that can learn dynamical systems from data in one call, given training data of trajectories (list of arrays), ```python import matplotlib.pyplot as plt import numpy as np

this is the convenience function

from autokoopman import auto_koopman

np.random.seed(20)

for a complete example, let's create an example dataset using an included benchmark system

import autokoopman.benchmark.fhn as fhn fhn = fhn.FitzHughNagumo() trainingdata = fhn.solveivps( initialstates=np.random.uniform(low=-2.0, high=2.0, size=(10, 2)), tspan=[0.0, 10.0], samplingperiod=0.1 )

learn model from data

experimentresults = autokoopman( trainingdata, # list of trajectories samplingperiod=0.1, # sampling period of trajectory snapshots obstype="rff", # use Random Fourier Features Observables opt="grid", # grid search to find best hyperparameters nobs=200, # maximum number of observables to try maxoptiter=200, # maximum number of optimization iterations gridparamslices=5, # for grid search, number of slices for each parameter n_splits=5, # k-folds validation for tuning, helps stabilize the scoring rank=(1, 200, 40) # rank range (start, stop, step) DMD hyperparameter )

get the model from the experiment results

model = experimentresults['tunedmodel']

simulate using the learned model

iv = [0.5, 0.1] trajectory = model.solveivp( initialstate=iv, tspan=(0.0, 10.0), sampling_period=0.1 )

simulate the ground truth for comparison

truetrajectory = fhn.solveivp( initialstate=iv, tspan=(0.0, 10.0), samplingperiod=0.1 )

plot the results

plt.plot(trajectory.states.T) plt.plot(true_trajectory.states.T) ```

Architecture

The library architecture has a modular design, allowing users to implement custom modules and plug them into the learning pipeline with ease.

AutoKoopman Class Structure in the Training Pipeline. A user can implement any of the classes to extend AutoKoopman (e.g., custom observables, a custom tuner, a new system id estimator).

Documentation

See the AutoKoopman Documentation.

Citing AutoKoopman

AutoKoopman has been published as a tool paper at ATVA 2023. The preprint can be found here.

If you cite AutoKoopman, please cite

Lew, E., Hekal, A., Potomkin, K., Kochdumper, N., Hencey, B., Bak, S., & Bogomolov, S. (2023, October). Autokoopman: A toolbox for automated system identification via koopman operator linearization. In International Symposium on Automated Technology for Verification and Analysis (pp. 237-250). Cham: Springer Nature Switzerland.

Bibtex: @inproceedings{lew2023autokoopman, title={Autokoopman: A toolbox for automated system identification via koopman operator linearization}, author={Lew, Ethan and Hekal, Abdelrahman and Potomkin, Kostiantyn and Kochdumper, Niklas and Hencey, Brandon and Bak, Stanley and Bogomolov, Sergiy}, booktitle={International Symposium on Automated Technology for Verification and Analysis}, pages={237--250}, year={2023}, organization={Springer} }

References

[1] Williams, M. O., Kevrekidis, I. G., & Rowley, C. W. (2015). A data–driven approximation of the koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25, 1307-1346.

[2] Li, Y., He, H., Wu, J., Katabi, D., & Torralba, A. (2019). Learning compositional koopman operators for model-based control. arXiv preprint arXiv:1910.08264.

[3] Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 113(15), 3932-3937.

[4] Bak, S., Bogomolov, S., Hencey, B., Kochdumper, N., Lew, E., & Potomkin, K. (2022, August). Reachability of Koopman linearized systems using random fourier feature observables and polynomial zonotope refinement. In Computer Aided Verification: 34th International Conference, CAV 2022, Haifa, Israel, August 7–10, 2022, Proceedings, Part I (pp. 490-510). Cham: Springer International Publishing.

[5] Proctor, J. L., Brunton, S. L., & Kutz, J. N. (2018). Generalizing Koopman theory to allow for inputs and control. SIAM Journal on Applied Dynamical Systems, 17(1), 909-930.

[6] Zhang, H., Rowley, C. W., Deem, E. A., & Cattafesta, L. N. (2019). Online dynamic mode decomposition for time-varying systems. SIAM Journal on Applied Dynamical Systems, 18(3), 1586-1609.