Operator inference for datadriven, nonintrusive model reduction of dynamical systems.
Project description
Operator Inference in Python
This is a Python implementation of Operator Inference for learning projectionbased polynomial reducedorder models of dynamical systems. The procedure is datadriven and nonintrusive, making it a viable candidate for model reduction of "glassbox" systems. The methodology was introduced in [1].
See the Wiki for mathematical details and API documentation. See this repository for a MATLAB implementation.
Quick Start
Installation
Install the package from the command line with the following single command (requires pip
).
$ python3 m pip install user romoperatorinference
See the wiki for other installation options.
Usage
Given a basis matrix Vr
, snapshot data X
, and snapshot time derivatives Xdot
, the following code learns a reduced model for a problem of the form dx / dt = c + Ax(t), then solves the reduced system for 0 ≤ t ≤ 1.
import numpy as np import rom_operator_inference as roi # Define a model of the form dx / dt = c + Ax(t). >>> model = roi.InferredContinuousROM(modelform="cA") # Fit the model to snapshot data X, the time derivatives Xdot, # and the linear basis Vr by solving for the operators c_ and A_. >>> model.fit(Vr, X, Xdot) # Simulate the learned model over the time domain [0,1] with 100 timesteps. >>> t = np.linspace(0, 1, 100) >>> x_ROM = model.predict(X[:,0], t)
Examples
The examples/
folder contains scripts and notebooks that set up and run several examples:
examples/tutorial.ipynb
: A walkthrough of a very simple heat equation example.examples/heat_1D.ipynb
: A more complicated onedimensional heat equation example [1].examples/data_driven_heat.ipynb
: A purely datadriven example using data generated from a onedimensional heat equation [4].
Contributors: Renee Swischuk, Shane McQuarrie, Elizabeth Qian, Boris Kramer, Karen Willcox.
References
These publications introduce, build on, or use Operator Inference. Entries are listed chronologically.

[1] Peherstorfer, B. and Willcox, K., Datadriven operator inference for nonintrusive projectionbased model reduction. Computer Methods in Applied Mechanics and Engineering, Vol. 306, pp. 196215, 2016. (Download)
BibTeX
@article{PW2016OperatorInference, title = {Datadriven operator inference for nonintrusive projectionbased model reduction}, author = {Peherstorfer, B. and Willcox, K.}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {306}, pages = {196215}, year = {2016}, publisher = {Elsevier} }

[2] Qian, E., Kramer, B., Marques, A., and Willcox, K., Transform & Learn: A datadriven approach to nonlinear model reduction. In the AIAA Aviation 2019 Forum & Exhibition, Dallas, TX, June 2019. Paper AIAA20193707. (Download)
BibTeX
@inbook{QKMW2019TransformAndLearn, title = {Transform \& Learn: A datadriven approach to nonlinear model reduction}, author = {Qian, E. and Kramer, B. and Marques, A. N. and Willcox, K. E.}, booktitle = {AIAA Aviation 2019 Forum}, year = {2018}, address = {Dallas, TX}, note = {Paper AIAA20193707}, doi = {10.2514/6.20193707}, URL = {https://arc.aiaa.org/doi/abs/10.2514/6.20193707}, eprint = {https://arc.aiaa.org/doi/pdf/10.2514/6.20193707} }

[3] Swischuk, R., Mainini, L., Peherstorfer, B., and Willcox, K., Projectionbased model reduction: Formulations for physicsbased machine learning. Computers & Fluids, Vol. 179, pp. 704717, 2019. (Download)
BibTeX
@article{SMPW2019PhysicsbasedML, title = {Projectionbased model reduction: Formulations for physicsbased machine learning}, author = {Swischuk, R. and Mainini, L. and Peherstorfer, B. and Willcox, K.}, journal = {Computers \& Fluids}, volume = {179}, pages = {704717}, year = {2019}, publisher = {Elsevier} }

[4] Swischuk, R., Physicsbased machine learning and datadriven reducedorder modeling. Master's thesis, Massachusetts Institute of Technology, 2019. (Download)
BibTeX
@phdthesis{swischuk2019MLandDDROM, title = {Physicsbased machine learning and datadriven reducedorder modeling}, author = {Swischuk, Renee}, year = {2019}, school = {Massachusetts Institute of Technology} }

[5] Peherstorfer, B. Sampling lowdimensional Markovian dynamics for preasymptotically recovering reduced models from data with operator inference. arXiv:1908.11233. (Download)
BibTeX
@article{peherstorfer2019samplingMarkovian, title = {Sampling lowdimensional Markovian dynamics for preasymptotically recovering reduced models from data with operator inference}, author = {Peherstorfer, Benjamin}, journal = {arXiv preprint arXiv:1908.11233}, year = {2019} }

[6] Swischuk, R., Kramer, B., Huang, C., and Willcox, K., Learning physicsbased reducedorder models for a singleinjector combustion process. AIAA Journal, Vol. 58:6, pp. 26582672, 2020. Also in Proceedings of 2020 AIAA SciTech Forum & Exhibition, Orlando FL, January, 2020. Paper AIAA20201411. Also Oden Institute Report 1913. (Download)
BibTeX
@article{SKHW2020ROMCombustion, title = {Learning physicsbased reducedorder models for a singleinjector combustion process}, author = {Swischuk, R. and Kramer, B. and Huang, C. and Willcox, K.}, journal = {AIAA Journal}, volume = {58}, number = {6}, pages = {26582672}, year = {2020}, publisher = {American Institute of Aeronautics and Astronautics} }

[7] Qian, E., Kramer, B., Peherstorfer, B., and Willcox, K., Lift & Learn: Physicsinformed machine learning for largescale nonlinear dynamical systems. Physica D: Nonlinear Phenomena, Vol. 406, May 2020, 132401. (Download)
BibTeX
@article{QKPW2020LiftAndLearn, title = {Lift \& Learn: Physicsinformed machine learning for largescale nonlinear dynamical systems.}, author = {Qian, E. and Kramer, B. and Peherstorfer, B. and Willcox, K.}, journal = {Physica {D}: {N}onlinear {P}henomena}, volume = {406}, pages = {132401}, url = {https://doi.org/10.1016/j.physd.2020.132401}, year = {2020} }

[8] Benner, P., Goyal, P., Kramer, B., Peherstorfer, B., and Willcox, K., Operator inference for nonintrusive model reduction of systems with nonpolynomial nonlinear terms. arXiv:2002.09726. Also Oden Institute Report 2004. (Download)
BibTeX
@article{BGKPW2020OpInfNonPoly, title = {Operator inference for nonintrusive model reduction of systems with nonpolynomial nonlinear terms}, author = {Benner, P. and Goyal, P. and Kramer, B. and Peherstorfer, B. and Willcox, K.}, journal = {arXiv preprint arXiv:2002.09726}, year = {2020} }

[9] Yıldız, S., Goyal, P., Benner, P., and Karasözen, B., Datadriven learning of reducedorder dynamics for a parametrized shallow water equation. arXiv:2007.14079. (Download)
BibTeX
@article{SGBK2020OpInfAffine, title = {DataDriven Learning of Reducedorder Dynamics for a Parametrized Shallow Water Equation}, author = {Y{\i}ld{\i}z, S. and Goyal, P. and Benner, P. and Karas{\"o}zen, B.}, journal = {arXiv preprint arXiv:2007.14079}, year = {2020} }

[10] McQuarrie, S. A., Huang, C., and Willcox, K., Datadriven reducedorder models via regularized operator inference for a singleinjector combustion process. arXiv:2008.02862. (Download)
BibTeX
@article{MHW2020regOpInfCombustion, title = {Datadriven reducedorder models via regularized operator inference for a singleinjector combustion process}, author = {McQuarrie, S. A. and Huang, C. and Willcox, K.}, journal = {arXiv preprint arXiv:2008.02862}, year = {2020} }
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