This is a Python implementation of Operator Inference for learning projection-based polynomial reduced-order models of dynamical systems. The procedure is data-driven and non-intrusive, making it a viable candidate for model reduction of "glass-box" systems. The methodology was introduced in [1].
See the Wiki for mathematical details and API documentation. See this repository for a MATLAB implementation.
Install the package from the command line with the following single command (requires pip).
$ python3 -m pip install --user rom-operator-inferenceSee the wiki for other installation options.
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)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 one-dimensional heat equation example [1].examples/data_driven_heat.ipynb: A purely data-driven example using data generated from a one-dimensional heat equation [4].
Contributors: Renee Swischuk, Shane McQuarrie, Elizabeth Qian, Boris Kramer, Karen Willcox.
These publications introduce, build on, or use Operator Inference. Entries are listed chronologically.
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[1] Peherstorfer, B. and Willcox, K., Data-driven operator inference for non-intrusive projection-based model reduction. Computer Methods in Applied Mechanics and Engineering, Vol. 306, pp. 196-215, 2016. (Download)
BibTeX
@article{PW2016OperatorInference, title = {Data-driven operator inference for nonintrusive projection-based model reduction}, author = {Peherstorfer, B. and Willcox, K.}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {306}, pages = {196--215}, year = {2016}, publisher = {Elsevier} } -
[2] Qian, E., Kramer, B., Marques, A., and Willcox, K., Transform & Learn: A data-driven approach to nonlinear model reduction. In the AIAA Aviation 2019 Forum & Exhibition, Dallas, TX, June 2019. Paper AIAA-2019-3707. (Download)
BibTeX
@inbook{QKMW2019TransformAndLearn, title = {Transform \& Learn: A data-driven 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 AIAA-2019-3707}, doi = {10.2514/6.2019-3707}, URL = {https://arc.aiaa.org/doi/abs/10.2514/6.2019-3707}, eprint = {https://arc.aiaa.org/doi/pdf/10.2514/6.2019-3707} } -
[3] Swischuk, R., Mainini, L., Peherstorfer, B., and Willcox, K., Projection-based model reduction: Formulations for physics-based machine learning. Computers & Fluids, Vol. 179, pp. 704-717, 2019. (Download)
BibTeX
@article{SMPW2019PhysicsbasedML, title = {Projection-based model reduction: Formulations for physics-based machine learning}, author = {Swischuk, R. and Mainini, L. and Peherstorfer, B. and Willcox, K.}, journal = {Computers \& Fluids}, volume = {179}, pages = {704--717}, year = {2019}, publisher = {Elsevier} } -
[4] Swischuk, R., Physics-based machine learning and data-driven reduced-order modeling. Master's thesis, Massachusetts Institute of Technology, 2019. (Download)
BibTeX
@phdthesis{swischuk2019MLandDDROM, title = {Physics-based machine learning and data-driven reduced-order modeling}, author = {Swischuk, Renee}, year = {2019}, school = {Massachusetts Institute of Technology} } -
[5] Peherstorfer, B. Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference. arXiv:1908.11233. (Download)
BibTeX
@article{peherstorfer2019samplingMarkovian, title = {Sampling low-dimensional Markovian dynamics for pre-asymptotically 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 physics-based reduced-order models for a single-injector combustion process. AIAA Journal, Vol. 58:6, pp. 2658-2672, 2020. Also in Proceedings of 2020 AIAA SciTech Forum & Exhibition, Orlando FL, January, 2020. Paper AIAA-2020-1411. Also Oden Institute Report 19-13. (Download)
BibTeX
@article{SKHW2020ROMCombustion, title = {Learning physics-based reduced-order models for a single-injector combustion process}, author = {Swischuk, R. and Kramer, B. and Huang, C. and Willcox, K.}, journal = {AIAA Journal}, volume = {58}, number = {6}, pages = {2658--2672}, year = {2020}, publisher = {American Institute of Aeronautics and Astronautics} } -
[7] Qian, E., Kramer, B., Peherstorfer, B., and Willcox, K., Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. Physica D: Nonlinear Phenomena, Vol. 406, May 2020, 132401. (Download)
BibTeX
@article{QKPW2020LiftAndLearn, title = {Lift \& Learn: Physics-informed machine learning for large-scale 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 non-intrusive model reduction of systems with non-polynomial nonlinear terms. arXiv:2002.09726. Also Oden Institute Report 20-04. (Download)
BibTeX
@article{BGKPW2020OpInfNonPoly, title = {Operator inference for non-intrusive model reduction of systems with non-polynomial 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., Data-driven learning of reduced-order dynamics for a parametrized shallow water equation. arXiv:2007.14079. (Download)
BibTeX
@article{SGBK2020OpInfAffine, title = {Data-Driven Learning of Reduced-order 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., Data-driven reduced-order models via regularized operator inference for a single-injector combustion process. arXiv:2008.02862. (Download)
BibTeX
@article{MHW2020regOpInfCombustion, title = {Data-driven reduced-order models via regularized operator inference for a single-injector combustion process}, author = {McQuarrie, S. A. and Huang, C. and Willcox, K.}, journal = {arXiv preprint arXiv:2008.02862}, year = {2020} }