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Cited by 171Model Reduction and Neural NetworksControl Systems and IdentificationChaos control and synchronization

Discovery of nonlinear multiscale systems: Sampling strategies and embeddings

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Champion, Kathleen P., Brunton, Steven L., Kutz, J. Nathan·2019·SIAM Journal on Applied Dynamical Systems
DOI 10.1137/18m1188227
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Abstract

A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. This problem is made more difficult by the fact that many systems of interest exhibit diverse behaviors across multiple time scales. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data, provided that accurate measurements of the derivatives can be computed from the data. We introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales; specifically, we can discover distinct governing equations at slow and fast scales. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates, can be used to obtain a linear model and Koopman invariant measurement system that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems on the attractor. We introduce two strategies for using HAVOK on systems with multiple time scales. Together, our approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.

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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 [23]

    The Abstract (p. 1) and Discussion (p. 17) describe SINDy and HAVOK methods for discovering dynamical systems, not applications of Variational System Identification to the listed subjects. Searches for variational identification, materials, and COVID-19 found no passage in our copy addressing those applications. (LLM, conf=0.77)

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References 52 · Cited by 171

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Authors
Champion, K. P., Brunton, S. L., & Kutz, J. N.
Affiliations
University of Washington (US)
ORCID
Venue
SIAM Journal on Applied Dynamical Systems
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Published
2019-01
Volume
18
Issue
1
Pages
312-333
ISSN
1536-0040 (ISSN-L)
DOI
10.1137/18m1188227
OpenAlex
W2803477776
Cited by
171
References
52
Citations by year
2026: 9 · 2025: 23 · 2024: 31 · 2023: 29 · 2022: 18 · 2021: 26 · 2020: 24 · 2019: 10 · 2018: 1
FWCI
9.03
Type
journal article
Language
en
Open access
Locations
  • SIAM Journal on Applied Dynamical Systems (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Defense Advanced Research Projects Agency (HR011-16-C-0016)
  • Air Force Office of Scientific Research (FA9550-17-1-0329)
  • Army Research Office (W911NF-17-1-0306)
  • National Science Foundation (DGE-1256082)
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 › Physics and Astronomy › Statistical and Nonlinear Physics › Chaos control and synchronization
Keywords
model discovery, sparse identification of nonlinear dynamical systems, SINDy, Koopman operator, time-delay embedding, incomplete measurements, sampling strategy, quasiperiodic systems, multiscale dynamics, data-driven modeling

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