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Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition
Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable
1. | Faculty of Marine Technology, Tokyo University of Marine Science and Technology, 2-1-6, Ecchujima, Koto-ku, Tokyo, 135-8533, Japan |
2. | Graduate School of Business Administration, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan |
3. | JST PRESTO, 4-1-8 Honcho, Kawaguchi-shi, Saitama 332-0012, Japan |
4. | Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA |
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.
References:
[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).
doi: 10.1103/PhysRevE.98.012215. |
[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).
doi: 10.1103/PhysRevFluids.3.104604. |
[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.
doi: 10.1063/1.5010285. |
[4] |
M. Inubushi and K. Yoshimura, Reservoir computing beyond memory-nonlinearity trade-off, Scientific Reports, 7 (2017).
doi: 10.1038/s41598-017-10257-6. |
[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.
doi: 10.1007/978-4-431-67002-5_11. |
[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 and H. Haas,
Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science, 304 (2004), 78-80.
doi: 10.1126/science.1091277. |
[9] |
Z. Lu, B. R. Hunt and E. Ott, Attractor reconstruction by machine learning, Chaos, 28 (2018), 9pp.
doi: 10.1063/1.5039508. |
[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).
doi: 10.1063/1.4979665. |
[11] |
M. Lukosevivcius and H. Jaeger,
Reservoir computing approaches to recurrent neural network training, Comput. Science Rev., 3 (2009), 127-149.
doi: 10.1016/j.cosrev.2009.03.005. |
[12] |
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 (2002), 2531-2560.
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).
doi: 10.1103/PhysRevE.98.023111. |
[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).
doi: 10.1103/PhysRevLett.120.024102. |
[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.
doi: 10.1063/1.5010300. |
[16] |
T. Sauer, J. A. Yorke and M. Casdagli,
Embedology, J. Statist. Phys., 65 (1991), 579-616.
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.
doi: 10.1007/BFb0091924. |
[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. |
[19] |
D. Verstraeten, B. Schrauwen, M. D'Haene and and D. A. Stroobandt,
An experimental unification of reservoir computing methods, Neural Network, 20 (2007), 391-403.
doi: 10.1016/j.neunet.2007.04.003. |
show all references
References:
[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).
doi: 10.1103/PhysRevE.98.012215. |
[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).
doi: 10.1103/PhysRevFluids.3.104604. |
[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.
doi: 10.1063/1.5010285. |
[4] |
M. Inubushi and K. Yoshimura, Reservoir computing beyond memory-nonlinearity trade-off, Scientific Reports, 7 (2017).
doi: 10.1038/s41598-017-10257-6. |
[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.
doi: 10.1007/978-4-431-67002-5_11. |
[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 and H. Haas,
Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science, 304 (2004), 78-80.
doi: 10.1126/science.1091277. |
[9] |
Z. Lu, B. R. Hunt and E. Ott, Attractor reconstruction by machine learning, Chaos, 28 (2018), 9pp.
doi: 10.1063/1.5039508. |
[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).
doi: 10.1063/1.4979665. |
[11] |
M. Lukosevivcius and H. Jaeger,
Reservoir computing approaches to recurrent neural network training, Comput. Science Rev., 3 (2009), 127-149.
doi: 10.1016/j.cosrev.2009.03.005. |
[12] |
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 (2002), 2531-2560.
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).
doi: 10.1103/PhysRevE.98.023111. |
[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).
doi: 10.1103/PhysRevLett.120.024102. |
[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.
doi: 10.1063/1.5010300. |
[16] |
T. Sauer, J. A. Yorke and M. Casdagli,
Embedology, J. Statist. Phys., 65 (1991), 579-616.
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.
doi: 10.1007/BFb0091924. |
[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. |
[19] |
D. Verstraeten, B. Schrauwen, M. D'Haene and and D. A. Stroobandt,
An experimental unification of reservoir computing methods, Neural Network, 20 (2007), 391-403.
doi: 10.1016/j.neunet.2007.04.003. |







variable | |
input variable | |
reservoir state vector | |
actual output variable obtained from Navier–Stokes equation | |
inferred output variable obtained from reservoir computing | |
weighted adjacency matrix | |
linear input weight | |
matrix used for translation from |
|
vector used for translation from |
|
normalized variable of |
variable | |
input variable | |
reservoir state vector | |
actual output variable obtained from Navier–Stokes equation | |
inferred output variable obtained from reservoir computing | |
weighted adjacency matrix | |
linear input weight | |
matrix used for translation from |
|
vector used for translation from |
|
normalized variable of |
parameter | Sec. 4 | Sec. 5 | |
dimension of input and output variables | 14 | Table. 3 | |
delay-time of the delay-coordinate | 4.0 | Table. 3 | |
dimension of reservoir state vector | 3000 | 2000 | |
parameter of determining |
120 | 80 | |
time step for reservoir dynamics | 0.5 | ||
transient time for |
3750 | ||
training time | 40000 | ||
number of iterations for the transient | 7500 | ||
number of iterations for the training | 80000 | ||
maximal eigenvalue of |
0.7 | ||
scale of input weights in |
0.5 | ||
nonlinearity degree of reservoir dynamics | 0.6 | ||
regularization parameter | 0.1 |
parameter | Sec. 4 | Sec. 5 | |
dimension of input and output variables | 14 | Table. 3 | |
delay-time of the delay-coordinate | 4.0 | Table. 3 | |
dimension of reservoir state vector | 3000 | 2000 | |
parameter of determining |
120 | 80 | |
time step for reservoir dynamics | 0.5 | ||
transient time for |
3750 | ||
training time | 40000 | ||
number of iterations for the transient | 7500 | ||
number of iterations for the training | 80000 | ||
maximal eigenvalue of |
0.7 | ||
scale of input weights in |
0.5 | ||
nonlinearity degree of reservoir dynamics | 0.6 | ||
regularization parameter | 0.1 |
| |||||||||||
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
3.0 | 0 | 0 | 0 | 0 | 0 | 1 | 19 | 24 | 43 | 37 | 27 |
3.5 | 0 | 0 | 0 | 11 | 20 | 28 | 57 | 48 | 21 | 11 | 7 |
4.0 | 0 | 3 | 18 | 43 | 107 | 59 | 21 | 14 | 2 | 4 | 5 |
4.5 | 3 | 14 | 43 | 54 | 21 | 15 | 8 | 1 | 1 | 1 | 0 |
5.0 | 10 | 24 | 26 | 19 | 9 | 1 | 1 | 1 | 0 | 0 | 0 |
| |||||||||||
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
3.0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 6 | 10 | 8 | 4 |
3.5 | 0 | 0 | 0 | 2 | 3 | 5 | 6 | 4 | 1 | 3 | 1 |
4.0 | 0 | 0 | 2 | 8 | 14 | 10 | 1 | 4 | 1 | 0 | 1 |
4.5 | 1 | 1 | 8 | 14 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
5.0 | 2 | 4 | 6 | 6 | 3 | 0 | 1 | 0 | 0 | 0 | 0 |
| |||||||||||
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
3.0 | 0 | 0 | 0 | 0 | 0 | 1 | 19 | 24 | 43 | 37 | 27 |
3.5 | 0 | 0 | 0 | 11 | 20 | 28 | 57 | 48 | 21 | 11 | 7 |
4.0 | 0 | 3 | 18 | 43 | 107 | 59 | 21 | 14 | 2 | 4 | 5 |
4.5 | 3 | 14 | 43 | 54 | 21 | 15 | 8 | 1 | 1 | 1 | 0 |
5.0 | 10 | 24 | 26 | 19 | 9 | 1 | 1 | 1 | 0 | 0 | 0 |
| |||||||||||
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
3.0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 6 | 10 | 8 | 4 |
3.5 | 0 | 0 | 0 | 2 | 3 | 5 | 6 | 4 | 1 | 3 | 1 |
4.0 | 0 | 0 | 2 | 8 | 14 | 10 | 1 | 4 | 1 | 0 | 1 |
4.5 | 1 | 1 | 8 | 14 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
5.0 | 2 | 4 | 6 | 6 | 3 | 0 | 1 | 0 | 0 | 0 | 0 |
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