Eur. Phys. J. Special Topics 223, 2665–2684 (2014)
https://doi.org/10.1140/epjst/e2014-02285-8

Review

Exploiting sparsity and equation-free architectures in complex systems

¹ Institute for Disease Modeling, Bellevue, WA 98004, USA
² Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
³ Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
⁴ Department of Biology, University of Washington, Seattle, WA 98195, USA

^a e-mail: joproctor@intven.com

Received: 3 July 2014
Revised: 17 October 2014
Published online: 10 December 2014

Abstract

Complex systems exhibit dynamics that typically evolve on low-dimensional attractors and may have sparse representation in some optimal basis. Recently developed compressive sensing techniques exploit this sparsity for state reconstruction and/or categorical identification from limited measurements. We argue that data-driven dimensionality reduction methods integrate naturally with sparse sensing in the context of complex systems. This framework works equally well with a physical model or in an equation-free context, where data is available but the governing equations may be unknown. We demonstrate the advantages of combining these methods on three prototypical examples: classification of dynamical regimes, optimal sensor placement, and equation-free dynamic model reduction. These examples motivate the potentially transformative role that state-of-the-art data methods and machine learning can play in the analysis of complex systems.