Skip to main content Accessibility help

Constrained sparse Galerkin regression

  • Jean-Christophe Loiseau (a1) and Steven L. Brunton (a2)


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.


Corresponding author

Email address for correspondence:


Hide All
Akaike, H. 1974 A new look at the statistical model identification. IEEE Trans. Autom. Control 19 (6), 716723.
Åkervik, E., Brandt, L., Henningson, D. S., Hœpffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18 (6), 068102.
Andersen, M. S., Dahl, J. & Vandenberghe, L.2013 CVXOPT: a Python package for convex optimization, version 1.1.6.
Bagheri, S. 2013 Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596623.
Bagheri, S., Brandt, L. & Henningson, D. S. 2009 Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263298.
Balajewicz, M. J., Dowell, E. H. & Noack, B. R. 2013 Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. J. Fluid Mech. 729, 285308.
Barbagallo, A., Sipp, D. & Schmid, P. J. 2009 Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641, 150.
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750756.
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215241.
Berkooz, G., Holmes, P. J. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1), 539575.
Billings, S. A. 2013 Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains. Wiley.
Bongard, J. & Lipson, H. 2007 Automated reverse engineering of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 104 (24), 99439948.
Brunton, S. L. & Noack, B. R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67 (5), 050801.
Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2016a Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113 (15), 39323937.
Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2016b Sparse identification of nonlinear dynamics with control (SINDYc). IFAC NOLCOS 49 (18), 710715.
Candès, E. J. 2006 Compressive sampling. In Proceedings of the International Congress of Mathematicians, vol. 3, pp. 14331452. European Mathematical Society.
Carini, M., Auteri, F. & Giannetti, F. 2015 Centre-manifold reduction of bifurcating flows. J. Fluid Mech. 767, 109145.
Carlberg, K., Barone, M. & Antil, H. 2017 Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction. J. Comput. Phys. 330, 693734.
Carlberg, K., Tuminaro, R. & Boggs, P. 2015 Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37 (2), B153B184.
Chartrand, R. 2011 Numerical differentiation of noisy, nonsmooth data. ISRN Appl. Math. 2011, 164564.
Donoho, D. L. 2006 Compressed sensing. IEEE Trans. Inform. Theory 52 (4), 12891306.
Fabbiane, N., Semeraro, O., Bagheri, S. & Henningson, D. S. 2014 Adaptive and model-based control theory applied to convectively unstable flows. Appl. Mech. Rev. 66 (6), 060801.
Fischer, P. F., Lottes, J. W. & Kerkemeir, S. G.2008 NEK5000: a fast and scalable high-order solver for computational fluid dynamics.
Glaz, B., Liu, L. & Friedmann, P. P. 2010 Reduced-order nonlinear unsteady aerodynamic modeling using a surrogate-based recurrence framework. AIAA J. 48 (10), 24182429.
Gloerfelt, X. 2008 Compressible proper orthogonal decomposition/Galerkin reduced-order model of self-sustained oscillations in a cavity. Phys. Fluids 20 (11), 115105.
Golub, G. H. & Van Loan, C. F. 2012 Matrix Computations, vol. 3. JHU Press.
Haken, H. 1983 Springer Series in Synergetics (ed. Cardona, M., Fulde, P. & Queisser, H.-J.), p. 269. Springer.
Holmes, P. J., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.
Ilak, M. & Rowley, C. W. 2008 Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20, 034103.
Illingworth, S. J., Morgans, A. S. & Rowley, C. W. 2010 Feedback control of flow resonances using balanced reduced-order models. J. Sound Vib. 330 (8), 15671581.
Johnson, S. G.2014 The NLopt nonlinear-optimization package.
Jones, E., Oliphant, T., Peterson, P. et al. 2001 SciPy: open source scientific tools for Python.
Juang, J.-N. & Pappa, R. S. 1985 An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid., Control Dyn. 8 (5), 620627.
Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M., Daviller, G., Osth, J., Krajnovic, S. & Niven, R. K. 2014 Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365414.
Krizhevsky, A., Sutskever, I. & Hinton, G. E. 2012 Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems 25 (ed. Pereira, F., Burges, C. J. C., Bottou, L. & Weinberger, K. Q.), pp. 10971105. Curran Associates.
Kukreja, S. L. & Brenner, M. J. 2007 Nonlinear system identification of aeroelastic systems: a structure-detection approach. In Identification and Control, pp. 117145. Springer.
Kukreja, S. L., Löfberg, J. & Brenner, M. J. 2006 A least absolute shrinkage and selection operator (lasso) for nonlinear system identification. IFAC Proc. 39 (1), 814819.
Kutz, J. N. 2017 Deep learning in fluid dynamics. J. Fluid Mech. 814, 14.
Kutz, J. N., Brunton, S. L., Brunton, B. W. & Proctor, J. L. 2016 Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems. SIAM.
Lee, C., Kim, J., Babcock, D. & Goodman, R. 1997 Application of neural networks to turbulence control for drag reduction. Phys. Fluids 9 (6), 17401747.
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155166.
Linscott, R. & Wiklund, T.2014 Parsimonious dynamical systems using the LASSO and the bootstrap.
Majda, A. J. & Harlim, J. 2012 Physics constrained nonlinear regression models for time series. Nonlinearity 26 (1), 201.
Mangan, N. M., Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2016 Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Trans. Mol. Biol. Multi-Scale Commun. 2 (1), 5263.
Mangan, N. M., Kutz, J. N., Brunton, S. L. & Proctor, J. L. 2017 Model selection for dynamical systems via sparse regression and information criteria. Proc. R. Soc. Lond. A 473, 20170009.
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.
McConaghy, T. 2011 Ffx: Fast, scalable, deterministic symbolic regression technology. In Genetic Programming Theory and Practice IX, pp. 235260. Springer.
Meliga, P. 2017 Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description. J. Fluid Mech. 826, 503521.
Mezić, I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41 (1–3), 309325.
Mezić, I. 2013 Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357378.
Milano, M. & Koumoutsakos, P. 2002 Neural network modeling for near wall turbulent flow. J. Comput. Phys. 182 (1), 126.
Nair, A. G. & Taira, K. 2015 Network-theoretic approach to sparsified discrete vortex dynamics. J. Fluid Mech. 768, 549571.
Noack, B. R., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335363.
Noack, B. R., Morzynski, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control. vol. 528. Springer Science & Business Media.
Noack, B. R., Stankiewicz, W., Morzynski, M. & Schmid, P. J. 2016 Recursive dynamic mode decomposition of a transient cylinder wake. J. Fluid Mech. 809, 843872.
Rossiter, J. E.1964 Wind tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Tech. Rep. Ministry of Aviation; Royal Aircraft Establishment; RAE Farnborough.
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 9971013.
Rowley, C. W., Colonius, T. & Basu, A. J. 2002 On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities. J. Fluid Mech. 455, 315346.
Rowley, C. W. & Dawson, S. 2017 Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49, 387417.
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 645, 115127.
Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2017 Data-driven discovery of partial differential equations. Sci. Adv. 3, e1602614.
Schaeffer, H. 2017 Learning partial differential equations via data discovery and sparse optimization. Proc. R. Soc. Lond. A 473, 20160446.
Schlegel, M. & Noack, B. R. 2015 On long-term boundedness of galerkin models. J. Fluid Mech. 765, 325352.
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.
Schmidt, M. & Lipson, H. 2009 Distilling free-form natural laws from experimental data. Science 324 (5923), 8185.
Schumm, M., Eberhard, B. & Monkewitz, P. A. 1994 Self-excited oscillations in the wake of two-dimensional bluff bodies and their control. J. Fluid Mech. 271, 1753.
Schwarz, G. et al. 1978 Estimating the dimension of a model. Ann. Stat. 6 (2), 461464.
Semaan, R., Kumar, P., Burnazzi, M., Tissot, G., Cordier, L. & Noack, B. R. 2016 Reduced-order modelling of the flow around a high-lift configuration with unsteady coanda blowing. J. Fluid Mech. 800, 72110.
Semeraro, O., Lusseyran, F., Pastur, L. & Jordan, P. 2017 Qualitative dynamics of wavepackets in turbulent jets. Phys. Rev. Fluids 2, 094605.
Sengupta, T. K., Haider, S. I., Parvathi, M. K. & Pallavi, G. 2015 Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow past a cylinder. Phys. Rev. E 91 (4), 043303.
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333358.
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.
Sipp, D. & Schmid, P. J. 2016 Linear closed-loop control of fluid instabilities and noise-induced perturbations: a review of approaches and tools. Appl. Mech. Rev. 68 (2), 020801.
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Part I: coherent structures. Q. Appl. Maths 45 (3), 561571.
Tadmor, G., Lehmann, O., Noack, B. R. & Morzyński, M. 2010 Mean field representation of the natural and actuated cylinder wake. Phys. Fluids 22 (3), 034102.
Tibshirani, R. 1996 Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B 58, 267288.
Tu, J. H., Rowley, C. W., Luchtenburg, D. M., Brunton, S. L. & Kutz, J. N. 2014 On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1 (2), 391421.
Wang, W. X., Yang, R., Lai, Y. C., Kovanis, V. & Grebogi, C. 2011 Predicting catastrophes in nonlinear dynamical systems by compressive sensing. Phys. Rev. Lett. 106, 154101.
Wiggins, S. 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, vol. 2. Springer Science & Business Media.
Willcox, K. & Peraire, J. 2002 Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (11), 23232330.
Williams, M. O., Kevrekidis, I. G. & Rowley, C. W. 2015 A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25 (6), 13071346.
Yamouni, S., Sipp, D. & Jacquin, L. 2013 Interaction between feedback aeroacoustic and acoustic resonance mechanisms in a cavity flow: a global stability analysis. J. Fluid Mech. 717, 134165.
Yao, C. & Bollt, E. M. 2007 Modeling and nonlinear parameter estimation with Kronecker product representation for coupled oscillators and spatiotemporal systems. Phys. D 227 (1), 7899.
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21 (2), 155165.
Zhang, H.-Q., Fey, U., Noack, B. R., König, M. & Eckelmann, H. 1995 On the transition of the cylinder wake. Phys. Fluids 7 (4), 779794.
Zhang, W., Wang, B., Ye, Z. & Quan, J. 2012 Efficient method for limit cycle flutter analysis based on nonlinear aerodynamic reduced-order models. AIAA J. 50 (5), 10191028.
Zhang, Z. J. & Duraisamy, K. 2015 Machine learning methods for data-driven turbulence modeling. In 22nd AIAA Computational Fluid Dynamics Conference, p. 2460. American Institute of Aeronautics and Astronautics.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Constrained sparse Galerkin regression

  • Jean-Christophe Loiseau (a1) and Steven L. Brunton (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed