Data-Driven Methods in Fluid Dynamics: Sparse Classification from Experimental Data



This work explores the use of data-driven methods, including machine learning and sparse sampling, for systems in fluid dynamics. In particular, camera images of a transitional separation bubble are used with dimensionality reduction and supervised classification techniques to discriminate between an actuated and an unactuated flow. After classification is demonstrated on full-resolution image data, similar classification performance is obtained using heavily subsampled pixels from the images. Finally, a sparse sensor optimization based on compressed sensing is used to determine optimal pixel locations for accurate classification. With 5–10 specially selected sensors, the median cross-validated classification accuracy is ≥ 97 %, as opposed to a random set of 5–10 pixels, which results in classification accuracy of 70–80 %. The methods developed here apply broadly to high-dimensional data from fluid dynamics experiments. Relevant connections between sparse sampling and the representation of high-dimensional data in a low-rank feature space are discussed.


Flow visualization Reduced-order models Proper orthogonal decomposition Machine learning Classification Sparse sampling Compressed sensing 



We would like to thank Mark Glauser for valuable suggestions that have improved this work, especially encouraging us to elaborate on the connection to big data. We gratefully acknowledge discussions with Josh Proctor about sparsity methods in machine learning. SLB and ZB acknowledge generous support from the Department of Energy (DOE DE-EE0006785). SLB also acknowledges support from the Air Force Office of Scientific Research (FA9550-14-1-0398) and from the University of Washington Department of Mechanical Engineering. SLB and BWB acknowledge sponsorship by the UW eScience Institute as Data Science Fellows. EK, AS, and BRN acknowledge additional support by the ANR SepaCoDe (ANR-11-BS09-018) and ANR TUCOROM (ANR-10-CEXC-0015).


  1. 1.
    S.L. Brunton, B.R. Noack, Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67, 050801–1–050801–48 (2015)Google Scholar
  2. 2.
    R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 2 (Addison Wesley, Reading, 2013)zbMATHGoogle Scholar
  3. 3.
    W.K. George, Insight into the dynamics of coherent structures from a proper orthogonal decomposition, In Symposium on Near Wall Turbulence in Dubrovnik, (Dubrovnik, 1988)Google Scholar
  4. 4.
    M.N. Glauser, S.J. Leib, W.K. George, Coherent Structures in the Axisymmetric Turbulent Jet Mixing Layer. (Springer, Berlin, 1987)Google Scholar
  5. 5.
    G. Berkooz, P. Holmes, J.L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 23, 539–575 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P.J. Holmes, J.L. Lumley, G. Berkooz, C.W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge Monographs in Mechanics (Cambridge University Press, Cambridge, 2012)Google Scholar
  7. 7.
    G. Golub, W. Kahan, Calculating the singular values and pseudo-inverse of a matrix. J. Soc. Ind. Appl. Math. Ser. B Numer. Anal. 2 (2), 205–224 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    G.H. Golub, C. Reinsch, Singular value decomposition and least squares solutions. Numer. Math. 14, 403–420 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    L. Sirovich, Turbulence and the dynamics of coherent structures, Parts I–III. Q. Appl. Math. XLV (3), 561–590 (1987)Google Scholar
  10. 10.
    S. Skogestad, I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, 2 edn. (Wiley, Hoboken, 2005)zbMATHGoogle Scholar
  11. 11.
    C.M. Bishop et al., Pattern Recognition and Machine Learning, vol. 1 (Springer, New York, 2006)zbMATHGoogle Scholar
  12. 12.
    J.N. Kutz, Data-Driven Modeling & Scientific Computation: Methods for Complex Systems & Big Data (Oxford University Press, Oxford, 2013)zbMATHGoogle Scholar
  13. 13.
    D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52 (4), 1289–1306 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52 (2), 489–509 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    E.J. Candès, T. Tao, Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52 (12), 5406–5425 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    E.J. Candès, Compressive sensing, in Proceedings of the International Congress of Mathematics, 2006Google Scholar
  17. 17.
    R.G. Baraniuk, Compressive sensing. IEEE Signal Process. Mag. 24 (4), 118–120 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J.A. Tropp, A.C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53 (12), 4655–4666, (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Hastie, R. Tibshirani, J. Friedman, T. Hastie, J. Friedman, R. Tibshirani, The Elements of Statistical Learning, vol. 2 (Springer, Berlin, 2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    G. James, D. Witten, T. Hastie, R. Tibshirani, An Introduction to Statistical Learning (Springer, Berlin, 2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    R. Tibshirani, Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodological) 58, 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  22. 22.
    J. Wright, A. Yang, A. Ganesh, S. Sastry, Y. Ma, Robust face recognition via sparse representation. IEEE Trans. Pattern Anal. Mach. Intell. (PAMI) 31 (2), 210–227 (2009)Google Scholar
  23. 23.
    L. Clemmensen, T. Hastie, D. Witten, B. Ersbøll, Sparse discriminant analysis. Technometrics 53 (4), 406–413 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    B.W. Brunton, S.L. Brunton, J.L. Proctor, J.N. Kutz, Optimal sensor placement and enhanced sparsity for classification. arXiv preprint arXiv:1310.4217, 2013Google Scholar
  25. 25.
    D.P. Hart, High-speed PIV analysis using compressed image correlation. J. Fluids Eng. 120, 463–470 (1998)CrossRefGoogle Scholar
  26. 26.
    S. Petra, C. Schn orr, TomoPIV meets compressed sensing. Pure Math. Appl. 20 (1–2), 49–76 (2009)Google Scholar
  27. 27.
    C.E. Willert, M. Gharib, Digital particle image velocimetry. Exp. Fluids 10 (4), 181–193 (1991)CrossRefGoogle Scholar
  28. 28.
    E. Kaiser, B.R. Noack, L. Cordier, A. Spohn, M. Segond, M. Abel, G. Daviller, J. Östh, S. Krajnovic, R.K. Niven, Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365–414 (2014)CrossRefzbMATHGoogle Scholar
  29. 29.
    D. Amsallem, M.J. Zahr, C. Farhat, Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Methods Eng. 92 (10), 891–916 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    A.G. Nair, K. Taira, Network-theoretic approach to sparsified discrete vortex dynamics. J. Fluid Mech. 768, 549–571 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    J.R Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection, vol. 1 (MIT Press, Cambridge, 1992)Google Scholar
  32. 32.
    N. Gautier, J.-L. Aider, T. Duriez, B.R. Noack, M. Segond, M. Abel, Closed-loop separation control using machine learning. J. Fluid Mech. 770, 442–457 (2015)CrossRefGoogle Scholar
  33. 33.
    T. Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.-P. Bonnet, L. Cordier, B.R. Noack, M. Segond, M. Abel, N. Gautier, J.-L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, S.L. Brunton, Closed-loop control of experimental shear flows using machine learning, in AIAA Paper 2014–2219, 7th Flow Control Conference, 2014Google Scholar
  34. 34.
    V. Parezanovic, J.-C. Laurentie, T. Duriez, C. Fourment, J. Delville, J.-P. Bonnet, L. Cordier, B. R. Noack, M. Segond, M. Abel, T. Shaqarin, S.L. Brunton, Mixing layer manipulation experiment – from periodic forcing to machine learning closed-loop control. J. Flow Turbul. Combust. 94 (1), 155–173 (2015)CrossRefGoogle Scholar
  35. 35.
    H. Nyquist, Certain topics in telegraph transmission theory. Trans. AIEE 47, 617–644 (1928)Google Scholar
  36. 36.
    C.E. Shannon, A mathematical theory of communication. Bell Syst. Techn. J. 27 (3), 379–423 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    I. Bright, G. Lin, J.N. Kutz, Compressive sensing and machine learning strategies for characterizing the flow around a cylinder with limited pressure measurements. Phys. Fluids 25, 127102–1–127102–15 (2013)Google Scholar
  38. 38.
    Z. Bai, T. Wimalajeewa, Z. Berger, G. Wang, M. Glauser, P.K Varshney, Low-dimensional approach for reconstruction of airfoil data via compressive sensing. AIAA J. 53 (4), 920–933 (2014)Google Scholar
  39. 39.
    J.-L. Bourguignon, J.A. Tropp, A.S. Sharma, B.J. McKeon, Compact representation of wall-bounded turbulence using compressive sampling. Phys. Fluids (1994–present) 26 (1), 015109 (2014)Google Scholar
  40. 40.
    I. Bright, G. Lin, J.N. Kutz, Classification of spatio-temporal data via asynchronous sparse sampling: application to flow around a cylinder. arXiv:1506.00661, 2015Google Scholar
  41. 41.
    C.W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, D.S. Henningson, Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    P.J. Schmid, Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    J.H. Tu, C.W. Rowley, D.M. Luchtenburg, S.L. Brunton, J.N. Kutz, On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1 (2), 391–421 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    M.R. Jovanović, P.J. Schmid, J.W. Nichols, Low-rank and sparse dynamic mode decomposition. Center for Turbulence Research, 2012Google Scholar
  45. 45.
    S.L. Brunton, J.L. Proctor, J.N. Kutz, Compressive sampling and dynamic mode decomposition. arXiv preprint arXiv:1312.5186, 2014Google Scholar
  46. 46.
    J.H. Tu, C.W. Rowley, J.N. Kutz, J.K. Shang, Spectral analysis of fluid flows using sub-Nyquist-rate PIV data. Exp. Fluids 55 (9), 1–13 (2014)CrossRefGoogle Scholar
  47. 47.
    F. Gueniat, L. Mathelin, L. Pastur, A dynamic mode decomposition approach for large and arbitrarily sampled systems. Phys. Fluids 27 (2), 025113 (2015)Google Scholar
  48. 48.
    J. Gosek, J.N. Kutz, Dynamic mode decomposition for real-time background/foreground separation in video. (2013, submitted for publication).
  49. 49.
    M.O. Williams, C.W. Rowley, I.G. Kevrekidis, A kernel approach to data-driven Koopman spectral analysis. arXiv preprint arXiv:1411.2260, 2014Google Scholar
  50. 50.
    M.O. Williams, I.G. Kevrekidis, C.W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. arXiv:1408.4408, 2014Google Scholar
  51. 51.
    M.S. Hemati, M.O. Williams, C.W. Rowley, Dynamic mode decomposition for large and streaming datasets. Phys. Fluids 26 (11), 111701 (2014)Google Scholar
  52. 52.
    M.O. Williams, C.W. Rowley, I. Mezić, I.G. Kevrekidis, Data fusion via intrinsic dynamic variables: an application of data-driven Koopman spectral analysis. Europhys. Lett. 109 (4), 40007 (2015)Google Scholar
  53. 53.
    J.L. Proctor, S.L. Brunton, J.N. Kutz, Dynamic mode decomposition with control: Using state and input snapshots to discover dynamics. SIAM J. Appl. Dyn. Syst. 15 (1), 142–161 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    H. Schaeffer, R. Caflisch, C.D. Hauck, S. Osher, Sparse dynamics for partial differential equations. Proc. Natl. Acad. Sci. USA 110 (17), 6634–6639 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    A. Mackey, H. Schaeffer, S. Osher, On the compressive spectral method. Multiscale Model. Simul. 12 (4), 1800–1827 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    S.L. Brunton, J.H. Tu, I. Bright, J.N. Kutz, Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems. SIAM J. Appl. Dyn. Syst. 13 (4), 1716–1732, (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    J.L. Proctor, S.L. Brunton, B.W. Brunton, J.N. Kutz, Exploiting sparsity and equation-free architectures in complex systems (invited review). Eur. Phys. J. Spec. Top. 223 (13), 2665–2684 (2014)CrossRefGoogle Scholar
  58. 58.
    F. Sommer, Mehrfachlösungen bei laminaren Strömungen mit Druckinduzierter Ablösung: eine Kuspen-Katastrophe. VDI Fortschrittsbericht, Reihe 7, Nr. 206, VDI Verlag Düsseldorf (Dissertation Bochum), pp. 429–443, 1992Google Scholar
  59. 59.
    F.A. Schraub, S.J. Kline, J. Henry, P.W. Runstadler, A. Littell, Use of hydrogen bubbles for quantitative determination of time-dependent velocity fields in low-speed water flows. J. Fluids Eng. 87 (2), 429–444 (1965)Google Scholar
  60. 60.
    M.N. Glauser, W.K. George, Application of multipoint measurements for flow characterization. Exp. Thermal Fluid Sci. 5 (5), 617–632 (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of WashingtonSeattleUSA
  2. 2.Department of BiologyUniversity of WashingtonSeattleUSA
  3. 3.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  4. 4.Départment Fluides, Thermique, CombustionInstitut PPRIME, CNRS – Université de Poitiers – ENSMAPoiters CEDEXFrance
  5. 5.Institute PPRIMECNRS – Université de Poitiers – ENSMAPoitiers CEDEXFrance
  6. 6.LIMSI-CNRS, Rue John von NeumannCampus Universitaire d’OrsayOrsayFrance
  7. 7.Institut für StrömungsmechanikTechnische Universit’́at BraunschweigBraunschweigGermany

Personalised recommendations