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Constrained sparse Galerkin regression

Published online by Cambridge University Press:  10 January 2018

Jean-Christophe Loiseau Open the ORCID record for Jean-Christophe Loiseau [Opens in a new window]  and
Steven L. Brunton
Jean-Christophe Loiseau*
Affiliation:
Laboratoire DynFluid, Arts et Métiers ParisTech, 75013 Paris, France
Steven L. Brunton
Affiliation:
Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
*
†Email address for correspondence: loiseau.jc@gmail.com
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Abstract

The sparse identification of nonlinear dynamics (SINDy) is a recently proposed data-driven modelling framework that uses sparse regression techniques to identify nonlinear low-order models. With the goal of low-order models of a fluid flow, we combine this approach with dimensionality reduction techniques (e.g. proper orthogonal decomposition) and extend it to enforce physical constraints in the regression, e.g. energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a high-fidelity solver to project the Navier–Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed measurement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of such an approach is demonstrated on two canonical flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favourably against reduced-order models obtained from a standard Galerkin projection procedure. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 42 - 67
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Copyright
© 2018 Cambridge University Press 

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