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.

Kathleen P. Champion

Steven L. Brunton

J. Nathan Kutz

SIAM Journal on Applied Dynamical Systems

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS (SIADS) 发表关于动力系统的数学分析和建模及其在物理、工程、生命和社会科学中的应用的研究文章。 SIADS 仅以电子格式发布。

The SIAM Journal on Applied Dynamical Systems (SIADS) publishes research articles on the mathematical analysis and modeling of dynamical systems and their applications in physics, engineering, life sciences, and social sciences. SIADS is published exclusively in electronic format.

SIAM Journal on Applied Dynamical Systems (SIADS) 发表关于动力系统的数学分析、建模及其在物理、工程、生命科学和社会科学中的应用研究文章。SIADS 仅以电子格式发布。

SIAM J. Appl. Dyn. Syst.

Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings

It is shown 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.

Category	Quartile
数学	4
数学, 物理数学物理	3
数学, 应用数学	4

Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings

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Journal Info

Category	Quartile
PHYSICS, MATHEMATICAL	2