Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable. (English) Zbl 1467.76046
Summary: We construct a data-driven dynamical system model for a macroscopic variable the Reynolds number of a high-dimensionally chaotic fluid flow by training its scalar time-series data. We use a machine-learning approach, the reservoir computing for the construction of the model, and do not use the knowledge of a physical process of fluid dynamics in its procedure. It is confirmed that an inferred time-series obtained from the model approximates the actual one and that some characteristics of the chaotic invariant set mimic the actual ones. We investigate the appropriate choice of the delay-coordinate, especially the delay-time and the dimension, which enables us to construct a model having a relatively high-dimensional attractor with low computational costs.
MSC:
76M99 | Basic methods in fluid mechanics |
76D05 | Navier-Stokes equations for incompressible viscous fluids |
68T05 | Learning and adaptive systems in artificial intelligence |
65P20 | Numerical chaos |
37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
Keywords:
machine learning; reservoir flow; Navier-Stokes equations; chaotic invariant set; high-dimensional attractorReferences:
[1] | P. Antonik, M. Gulina, J. Pauwels and S. Massar, Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronization and cryptography, Phys. Rev. E, 98 (2018). |
[2] | P. C. Di Leoni, A. Mazzino and L. Biferale, Inferring flow parameters and turbulent configuration with physics-informed data assimilation and spectral nudging, Phys. Rev. Fluids, 3 (2018). |
[3] | D. Ibáñez-Soria, J. Garcia-Ojalvo, A. Soria-Frisch and G. Ruffini, Detection of generalized synchronization using echo state networks, Chaos, 28 (2018), 7pp. · Zbl 1390.34135 |
[4] | M. Inubushi and K. Yoshimura, Reservoir computing beyond memory-nonlinearity trade-off, Scientific Reports, 7 (2017). |
[5] | T. Ishihara and Y. Kaneda, High resolution DNS of incompressible homogeneous forced turbulence-time dependence of the statistics, in Statistical Theories and Computational Approaches to Turbulence, Springer, Tokyo, 2003,177-188. · Zbl 1082.76559 |
[6] | K. Ishioka, ispack-0.4.1, 1999. Available from: http://www.gfd-dennou.org/arch/ispack/. |
[7] | H. Jaeger, The “echo state” approach to analysing and training recurrent neural networks, GMD Report, 148 (2001). |
[8] | H. Jaeger; H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science, 304, 78-80 (2004) · doi:10.1126/science.1091277 |
[9] | Z. Lu, B. R. Hunt and E. Ott, Attractor reconstruction by machine learning, Chaos, 28 (2018), 9pp. |
[10] | 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 (2017). |
[11] | M. Lukosevivcius; H. Jaeger, Reservoir computing approaches to recurrent neural network training, Comput. Science Rev., 3, 127-149 (2009) · Zbl 1302.68235 · doi:10.1016/j.cosrev.2009.03.005 |
[12] | W. Maass; T. Natschläger; H. Markram, Real-time computing without stable states: A new framework for neural computation based on perturbations, Neural Comput., 14, 2531-2560 (2002) · Zbl 1057.68618 · doi:10.1162/089976602760407955 |
[13] | K. Nakai and Y. Saiki, Machine-learning inference of fluid variables from data using reservoir computing, Phys. Rev. E, 98 (2018). |
[14] | J. Pathak, B. Hunt, M. Girvan, Z. Lu and E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Phys. Rev. Lett., 120 (2018). |
[15] | J. Pathak, Z. Lu, B. Hunt, M. Girvan and E. Ott, Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data, Chaos, 27 (2017), 9pp. · Zbl 1390.37138 |
[16] | T. Sauer; J. A. Yorke; M. Casdagli, Embedology, J. Statist. Phys., 65, 579-616 (1991) · Zbl 0943.37506 · doi:10.1007/BF01053745 |
[17] | F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, Lecture Notes in Math., 898, Springer, Berlin-New York, 1981,366-381. · Zbl 0513.58032 |
[18] | A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. · Zbl 0354.65028 |
[19] | D. Verstraeten; B. Schrauwen; M. D’Haene; and D. A. Stroobandt, An experimental unification of reservoir computing methods, Neural Network, 20, 391-403 (2007) · Zbl 1132.68605 · doi:10.1016/j.neunet.2007.04.003 |
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