Operator Inference

This is a Python implementation of Operator Inference for constructing projection-based reduced-order models of dynamical systems with a polynomial form. The procedure is data-driven and non-intrusive, making it a viable candidate for model reduction of black-box or complex systems. The methodology was introduced in [1]. See References for more papers that use or build on Operator Inference.

See this repository for a MATLAB implementation and DOCUMENTATION.md for the code documentation.

Problem Statement

Consider the (possibly nonlinear) system of n ordinary differential equations with state variable x, input (control) variable u, and independent variable t:

where

This system is called the full-order model (FOM). If n is large, as it often is in high-consequence engineering applications, it is computationally expensive to numerically solve the FOM. This package provides tools for constructing a reduced-order model (ROM) that is up to quadratic in the state x with optional linear control inputs u. The procedure is data-driven, non-intrusive, and relatively inexpensive. In the most general case, the code can construct and solve a reduced-order system with the polynomial form

where now

This reduced low-dimensional system approximates the original high-dimensional system, but it is much easier (faster) to solve because of its low dimension r << n.

See DETAILS.md for more mathematical details and an index of notation.

Quick Start

Installation

Install from the command line with the following single command (requires pip).

$ pip3 install rom-operator-inference

Usage

Given a linear basis 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 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].

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Contributors: Renee Swischuk, Shane McQuarrie, Elizabeth Qian, Boris Kramer, Karen Willcox.

References

• [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. (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},
  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. 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}
  }