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Cited by 4,796Model Reduction and Neural NetworksControl Systems and IdentificationNumerical methods for differential equations

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

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Brunton, Steven L., Proctor, Joshua L., Kutz, J. Nathan·2016·Proceedings of the National Academy of Sciences
DOI 10.1073/pnas.1517384113PMC PMC4839439
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Abstract

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.

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  • Variational System Identification has been applied to identify governing equations for the evolution of materials, the spatiotemporal spread of COVID-19, and constitutive models of soft materials.

    “Variational System Identification has been applied to identify governing equations for the evolution of materials17,20,19, the spatiotemporal spread of COVID-1921,18, as well as constitutive models of soft materials22,23.”

    From Inference of weak-form partial differential equations describing migration and proliferation mechanisms in wound healing experiments on cancer cells (2025) · cites this paper as [19]

    Section 3 (p. 4) identifies the paper’s method as sparse identification of nonlinear dynamics (SINDy), not Variational System Identification. The Abstract (p. 1) lists demonstrations on oscillators, the Lorenz system, and fluid vortex shedding; it does not report the claimed applications to materials, COVID-19, or soft-material constitutive models. (LLM, conf=0.86)

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  • It is possible to venture into learning the fundamental physics by learning the underlying equations for well-known systems and the structure of parameterizations from data, with the advantage of intrinsic interpretability.

    “Ideally, it is possible to venture into learning the fundamental physics by learning the underlying equations for well-known systems (Brunton et al., 2016) and the structure of parameterizations from data, with the advantage of intrinsic interpretability (Zanna & Bolton, 2020).”

    From Artificial intelligence for climate prediction of extremes: State of the art, challenges, and future perspectives (2024) · cites this paper as Brunton et al., 2016

  • Variational System Identification extends SINDy to inferring PDEs in weak form, thereby offering advantages in the presence of noise and high derivatives.

    “It extends the popular SINDy approach12,13,14,15,16 to inferring PDEs in weak form, thereby offering advantages in the presence of noise and high derivatives17,18,19.”

    From Inference of weak-form partial differential equations describing migration and proliferation mechanisms in wound healing experiments on cancer cells (2025) · cites this paper as [19]

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Authors
Brunton, S. L., Proctor, J. L., & Kutz, J. N.
Affiliations
  • University of Washington (US)
  • Institute for Disease Modeling (US)
Corresponding author
Brunton, Steven L.
ORCID
Venue
Proceedings of the National Academy of Sciences
Publisher
National Academy of Sciences
Published
2016-03-28
Volume
113
Issue
15
Pages
3932-3937
ISSN
  • 0027-8424 (ISSN-L)
  • 1091-6490
DOI
10.1073/pnas.1517384113
PMC
PMC4839439
OpenAlex
W2239232218
Cited by
4,796
References
4
Citations by year
2026: 788 · 2025: 812 · 2024: 771 · 2023: 647 · 2022: 522 · 2021: 466 · 2020: 360 · 2019: 233 · 2018: 122 · 2017: 51 · 2016: 22 · 2015: 1
FWCI
81.01
Type
journal article
Language
en
Open access
Licence
www.pnas.org/preview_site/misc/userlicense.xhtml (vor)
Locations
  • Proceedings of the National Academy of Sciences (publishedVersion)
  • PubMed Central (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
DOD | Air Force Office of Scientific Research (FA9550-09-0174)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Model Reduction and Neural Networks
  • Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
  • Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
Keywords
sparse identification of nonlinear dynamical systems, sparse regression, nonlinear dynamical systems, parsimonious models, nonlinear oscillators, Lorenz system, vortex shedding, parameterized systems, time-varying systems, external forcing

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