Recent advances in machine learning (ML) have facilitated its application to a wide range of systems, from complex to quantum. Reservoir computing algorithms have proven particularly effective for studying nonlinear dynamical systems that exhibit collective behaviors, such as synchronizations and chaotic phenomena, some of which still remain unclear. Here, we apply ML approaches to the Kuramoto model to address several intriguing problems, including identifying the transition point and criticality of a hybrid synchronization transition, predicting future chaotic behaviors, and understanding network structures from chaotic patterns. Our proposed method also has further implications, such as inferring the structure of neural networks from electroencephalogram signals. This study, finally, highlights the potential of ML approaches for advancing our understanding of complex systems.
REFERENCES
1.
W.
Maass
, T.
Natschläger
, and H.
Markram
, Neural Comput.
14
, 2531
(2002
). 2.
H.
Jaeger
and H.
Haas
, Science
304
, 78
(2004
). 3.
M.
Lukoševičius
and H.
Jaeger
, “Reservoir computing approaches to recurrent neural network training,” Comput. Sci. Rev.
3
(3), 127–149
(2009
).4.
Z.
Lu
, J.
Pathak
, B.
Hunt
, M.
Girvan
, R.
Brockett
, and E.
Ott
, Chaos
27
, 041102
(2017
). 5.
T. L.
Carroll
, Phys. Rev. E
98
, 052209
(2018
). 6.
Z.
Lu
, B. R.
Hunt
, and E.
Ott
, Chaos
28
, 061104
(2018
). 7.
J.
Pathak
, Z.
Lu
, B. R.
Hunt
, M.
Girvan
, and E.
Ott
, Chaos
27
, 121102
(2017
). 8.
J.
Pathak
, B.
Hunt
, M.
Girvan
, Z.
Lu
, and E.
Ott
, Phys. Rev. Lett.
120
, 024102
(2018
). 9.
T.
Weng
, H.
Yang
, C.
Gu
, J.
Zhang
, and M.
Small
, Phys. Rev. E
99
, 042203
(2019
). 10.
J.
Jiang
and Y.-C.
Lai
, Phys. Rev. Res.
1
, 033056
(2019
). 11.
H.
Fan
, J.
Jiang
, C.
Zhang
, X.
Wang
, and Y.-C.
Lai
, Phys. Rev. Res.
2
, 012080
(2020
). 12.
J. Z.
Kim
, Z.
Lu
, E.
Nozari
, G. J.
Pappas
, and D. S.
Bassett
, Nat. Mach. Intell.
3
, 316
(2021
). 13.
M.
Nitzan
, J.
Casadiego
, and M.
Timme
, Sci. Adv.
3
, e1600396
(2017
). 14.
W.-X.
Wang
, Y.-C.
Lai
, and C.
Grebogi
, Phys. Rep.
644
, 1
(2016
). 15.
D.
Eroglu
, M.
Tanzi
, S.
van Strien
, and T.
Pereira
, “Revealing dynamics, communities, and criticality from data,” Phys. Rev. X
10
, 021047
(2020
).16.
F.
Mormann
, T.
Kreuz
, C.
Rieke
, R. G.
Andrzejak
, A.
Kraskov
, P.
David
, C. E.
Elger
, and K.
Lehnertz
, Clin. Neurophysiol.
116
, 569
(2005
). 17.
P.
Mirowski
, D.
Madhavan
, Y.
LeCun
, and R.
Kuzniecky
, Clin. Neurophysiol.
120
, 1927
(2009
). 18.
S.
Chandaka
, A.
Chatterjee
, and S.
Munshi
, Expert Syst. Appl.
36
, 1329
(2009
). 19.
J. R.
Williamson
, D. W.
Bliss
, D. W.
Browne
, and J. T.
Narayanan
, “Seizure prediction using EEG spatiotemporal correlation structure,” Epilepsy Behav.
25
(2), 230–238
(2012
).20.
J.
Carrasquilla
and R. G.
Melko
, Nat. Phys.
13
, 431
(2017
). 21.
P.
Broecker
, J.
Carrasquilla
, R. G.
Melko
, and S.
Trebst
, Sci. Rep.
7
, 1
(2017
). 22.
J.
Venderley
, V.
Khemani
, and E.-A.
Kim
, Phys. Rev. Lett.
120
, 257204
(2018
). 23.
M. J. S.
Beach
, A.
Golubeva
, and R. G.
Melko
, Phys. Rev. B
97
, 045207
(2018
). 24.
A.
Bohrdt
, C. S.
Chiu
, G.
Ji
, M.
Xu
, D.
Greif
, M.
Greiner
, E.
Demler
, F.
Grusdt
, and M.
Knap
, Nat. Phys.
15
, 921
(2019
). 25.
W.
Zhang
, J.
Liu
, and T.-C.
Wei
, Phys. Rev. E
99
, 032142
(2019
). 26.
W.
Yu
and P.
Lyu
, Phys. A
559
, 125065
(2020
). 27.
28.
X.
Zhong
and S.
Wang
, Symmetry
14
, 1084
(2022
). 29.
L.
Wang
, H.
Fan
, J.
Xiao
, Y.
Lan
, and X.
Wang
, “Criticality in reservoir computer of coupled phase oscillators,” Phys. Rev. E
105
, L052201
(2022
).30.
C.
Zhang
, J.
Jiang
, S.-X.
Qu
, and Y.-C.
Lai
, Chaos
30
, 083114
(2020
). 31.
Y.
Kuramoto
, International Symposium on Mathematical Problems in Theoretical Physics
(Springer-Verlag
, Berlin
, 1975
), pp. 420
–422
.32.
Y.
Kuramoto
, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, Vol. 19 (Springer, Berlin, 1984).33.
E. A.
Martens
, E.
Barreto
, S. H.
Strogatz
, E.
Ott
, P.
So
, and T. M.
Antonsen
, Phys. Rev. E
79
, 026204
(2009
). 34.
D.
Pazó
and E.
Montbrió
, “Existence of hysteresis in the Kuramoto model with bimodal frequency distributions,” Phys. Rev. E
80
, 046215
(2009
).35.
P. S.
Skardal
, Phys. Rev. E
98
, 022207
(2018
). 36.
L.-H.
Tang
, J. Stat. Mech. Theory Exp.
2011
, P01034
(2011
). 37.
F. A.
Rodrigues
, T. K. D.
Peron
, P.
Ji
, and J.
Kurths
, Phys. Rep.
610
, 1
(2016
). 38.
D.
Pazó
, “Thermodynamic limit of the first-order phase transition in the Kuramoto model,” Phys. Rev. E
72
, 046211
(2005
).39.
L.
Basnarkov
and V.
Urumov
, Phys. Rev. E
76
, 057201
(2007
). 40.
B. C.
Coutinho
, A. V.
Goltsev
, S. N.
Dorogovtsev
, and J. F. F.
Mendes
, Phys. Rev. E
87
, 032106
(2013
). 41.
J. U.
Song
, J.
Um
, J.
Park
, and B.
Kahng
, Phys. Rev. E
101
, 052313
(2020
). 42.
C.
Choi
, M.
Ha
, and B.
Kahng
, Phys. Rev. E
88
, 032126
(2013
). 43.
S.
Yoon
, M.
Sorbaro Sindaci
, A. V.
Goltsev
, and J. F. F.
Mendes
, Phys. Rev. E
91
, 032814
(2015
). 44.
R.
Rossi
and N.
Ahmed
, Proc. AAAI Conf. Artif. Intell.
29
, 4292–4293
(2015
). 45.
D. D.
Bock
, W.-C. A.
Lee
, A. M.
Kerlin
, M. L.
Andermann
, G.
Hood
, A. W.
Wetzel
, S.
Yurgenson
, E. R.
Soucy
, H. S.
Kim
, and R. C.
Reid
, Nature
471
, 177
(2011
). 46.
47.
H.
Hong
, H.
Park
, and M. Y.
Choi
, Phys. Rev. E
72
, 036217
(2005
). 48.
H.
Hong
, H.
Chaté
, H.
Park
, and L.-H.
Tang
, Phys. Rev. Lett.
99
, 184101
(2007
). 49.
J.
Um
, H.
Hong
, and H.
Park
, Phys. Rev. E
89
, 012810
(2014
). 50.
H.
Hong
, H.
Chaté
, L.-H.
Tang
, and H.
Park
, Phys. Rev. E
92
, 022122
(2015
). 51.
E.
Oh
, K.
Rho
, H.
Hong
, and B.
Kahng
, Phys. Rev. E
72
, 047101
(2005
). 52.
A.
Arenas
, A.
Díaz-Guilera
, and C. J.
Pérez-Vicente
, Phys. Rev. Lett.
96
, 114102
(2006
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
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