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Research Article

Discovering governing equations from data by sparse identification of nonlinear dynamical systems

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz
  1. aDepartment of Mechanical Engineering, University of Washington, Seattle, WA 98195;
  2. bInstitute for Disease Modeling, Bellevue, WA 98005;
  3. cDepartment of Applied Mathematics, University of Washington, Seattle, WA 98195

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PNAS April 12, 2016 113 (15) 3932-3937; first published March 28, 2016; https://doi.org/10.1073/pnas.1517384113
Steven L. Brunton
aDepartment of Mechanical Engineering, University of Washington, Seattle, WA 98195;
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  • For correspondence: sbrunton@uw.edu
Joshua L. Proctor
bInstitute for Disease Modeling, Bellevue, WA 98005;
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J. Nathan Kutz
cDepartment of Applied Mathematics, University of Washington, Seattle, WA 98195
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  1. Edited by William Bialek, Princeton University, Princeton, NJ, and approved March 1, 2016 (received for review August 31, 2015)

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Significance

Understanding dynamic constraints and balances in nature has facilitated rapid development of knowledge and enabled technology, including aircraft, combustion engines, satellites, and electrical power. This work develops a novel framework to discover governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity techniques and machine learning. The resulting models are parsimonious, balancing model complexity with descriptive ability while avoiding overfitting. There are many critical data-driven problems, such as understanding cognition from neural recordings, inferring climate patterns, determining stability of financial markets, predicting and suppressing the spread of disease, and controlling turbulence for greener transportation and energy. With abundant data and elusive laws, data-driven discovery of dynamics will continue to play an important role in these efforts.

Abstract

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

  • dynamical systems
  • machine learning
  • sparse regression
  • system identification
  • optimization

Footnotes

  • ↵1To whom correspondence should be addressed. Email: sbrunton{at}uw.edu.
  • Author contributions: S.L.B., J.L.P., and J.N.K. designed research; S.L.B. performed research; S.L.B., J.L.P., and J.N.K. analyzed data; and S.L.B. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1517384113/-/DCSupplemental.

Freely available online through the PNAS open access option.

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Sparse identification of nonlinear dynamics
Steven L. Brunton, Joshua L. Proctor, J. Nathan Kutz
Proceedings of the National Academy of Sciences Apr 2016, 113 (15) 3932-3937; DOI: 10.1073/pnas.1517384113

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Sparse identification of nonlinear dynamics
Steven L. Brunton, Joshua L. Proctor, J. Nathan Kutz
Proceedings of the National Academy of Sciences Apr 2016, 113 (15) 3932-3937; DOI: 10.1073/pnas.1517384113
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  • Physical Sciences
  • Applied Mathematics
Proceedings of the National Academy of Sciences: 113 (15)
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