The applicability of machine learning for predicting chaotic dynamics relies heavily upon the data used in the training stage. Chaotic time series obtained by numerically solving ordinary differential equations embed a complicated noise of the applied numerical scheme. Such a dependence of the solution on the numeric scheme leads to an inadequate representation of the real chaotic system. A stochastic approach for generating training time series and characterizing their predictability is suggested to address this problem. The approach is applied for analyzing two chaotic systems with known properties, the Lorenz system and the Anishchenko–Astakhov generator. Additionally, the approach is extended to critically assess a reservoir computing model used for chaotic time series prediction. Limitations of reservoir computing for surrogate modeling of chaotic systems are highlighted.

1.
R.
Lifshitz
and
M. C.
Cross
, “Nonlinear dynamics of nanomechanical and micromechanical resonators,” in Reviews of Nonlinear Dynamics and Complexity (John Wiley & Sons, Ltd., 2008), Chap. 1, pp. 1–52.
2.
H.
Jaeger
and
H.
Haas
, “
Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication
,”
Science
304
,
78
80
(
2004
).
3.
G.
Tanaka
,
T.
Yamane
,
J. B.
Heroux
,
R.
Nakane
,
N.
Kanazawa
,
S.
Takeda
,
H.
Numata
,
D.
Nakano
, and
A.
Hirose
, “
Recent advances in physical reservoir computing: A review
,”
Neural Netw.
115
,
100
123
(
2019
).
4.
L.
Appeltant
,
M. C.
Soriano
,
G.
Van der Sande
,
J.
Danckaert
,
S.
Massar
,
J.
Dambre
,
B.
Schrauwen
,
C. R.
Mirasso
, and
I.
Fischer
, “
Information processing using a single dynamical node as complex system
,”
Nat. Commun.
2
,
468
(
2011
).
5.
C.
Du
,
F.
Cai
,
M. A.
Zidan
,
W.
Ma
,
S. H.
Lee
, and
W. D.
Lu
, “
Reservoir computing using dynamic memristors for temporal information processing
,”
Nat. Commun.
8
,
2204
(
2017
).
6.
Y.
Zhong
,
J.
Tang
,
X.
Li
,
B.
Gao
,
H.
Qian
, and
H.
Wu
, “
Dynamic memristor-based reservoir computing for high-efficiency temporal signal processing
,”
Nat. Commun.
12
,
408
(
2021
).
7.
M.
Romera
,
P.
Talatchian
,
S.
Tsunegi
,
F.
Abreu Araujo
,
V.
Cros
,
P.
Bortolotti
,
J.
Trastoy
,
K.
Yakushiji
,
A.
Fukushima
,
H.
Kubota
,
S.
Yuasa
,
M.
Ernoult
,
D.
Vodenicarevic
,
T.
Hirtzlin
,
N.
Locatelli
,
D.
Querlioz
, and
J.
Grollier
, “
Vowel recognition with four coupled spin-torque nano-oscillators
,”
Nature
563
,
230
234
(
2018
).
8.
L.
Larger
,
A.
Baylón-Fuentes
,
R.
Martinenghi
,
V. S.
Udaltsov
,
Y. K.
Chembo
, and
M.
Jacquot
, “
High-speed photonic reservoir computing using a time-delay-based architecture: Million words per second classification
,”
Phys. Rev. X
7
,
011015
(
2017
).
9.
P.
Antonik
,
M.
Haelterman
, and
S.
Massar
, “
Brain-inspired photonic signal processor for generating periodic patterns and emulating chaotic systems
,”
Phys. Rev. Appl.
7
,
054014
(
2017
).
10.
G.
Dion
,
S.
Mejaouri
, and
J.
Sylvestre
, “
Reservoir computing with a single delay-coupled non-linear mechanical oscillator
,”
J. Appl. Phys.
124
,
152132
(
2018
).
11.
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
,
024102
(
2018
).
12.
R. S.
Zimmermann
and
U.
Parlitz
, “
Observing spatio-temporal dynamics of excitable media using reservoir computing
,”
Chaos
28
,
043118
(
2018
).
13.
H.
Fan
,
J.
Jiang
,
C.
Zhang
,
X.
Wang
, and
Y.-C.
Lai
, “
Long-term prediction of chaotic systems with machine learning
,”
Phys. Rev. Res.
2
,
012080
(
2020
).
14.
T.
Weng
,
H.
Yang
,
C.
Gu
,
J.
Zhang
, and
M.
Small
, “
Synchronization of chaotic systems and their machine-learning models
,”
Phys. Rev. E
99
,
042203
(
2019
).
15.
L.-W.
Kong
,
H.-W.
Fan
,
C.
Grebogi
, and
Y.-C.
Lai
, “
Machine learning prediction of critical transition and system collapse
,”
Phys. Rev. Res.
3
,
013090
(
2021
).
16.
M. U.
Kobayashi
,
K.
Nakai
,
Y.
Saiki
, and
N.
Tsutsumi
, “Dynamical system analysis of a data-driven model constructed by reservoir computing,” arXiv:2102.13475 (2021).
17.
E. A.
Jackson
,
Perspectives of Nonlinear Dynamics
(
Cambridge University Press
,
1990
), Vol. 2.
18.
S.
Gonchenko
,
L.
Shil’nikov
, and
D.
Turaev
, “
On models with non-rough Poincaré homoclinic curves
,”
Physica D
62
,
1
14
(
1993
).
19.
S. V.
Gonchenko
,
L. P.
Shilnikov
, and
D. V.
Turaev
, “
On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I
,”
Nonlinearity
21
,
923
972
(
2008
).
20.
E. N.
Lorenz
, “
Deterministic non-periodic flow
,”
J. Atmos. Sci.
20
,
130
141
(
1963
).
21.
V. S.
Anishchenko
,
V. V.
Astakhov
, and
T. E.
Letchford
, “
Multifrequency and stochastic autooscillations in the oscillator with inertia nonlinearity
,”
Radiotek. Elektron.
27
,
1972
1978
(
1982
).
22.
V. S.
Anishchenko
and
V. V.
Astakhov
, “
Experimental investigation of the mechanism of arising and the structure of a strange attractor in a generator with inertia nonlinearity
,”
Radiotek. Elektron.
28
,
1109
1115
(
1983
).
23.
V. S.
Anishchenko
,
Complex Oscillations in Simple Systems
(
Nauka
,
Moscow
,
1990
).
24.
J. C.
Butcher
,
Numerical Methods for Ordinary Differential Equations
(
John Wiley & Sons
,
Hoboken, NJ
,
2016
).
25.
M.
Lukoševičius
, “A practical guide to applying echo state networks,” in Neural Networks: Tricks of the Trade, 2nd ed., edited by G. Montavon, G. B. Orr, and K.-R. Müller (Springer, Berlin, 2012), pp. 659–686.
26.
V.
Arnold
,
V.
Afrajmovich
,
Y.
Il’yashenko
, and
L.
Shil’nikov
,
Dynamical Systems V
(
Springer-Verlag
,
Berlin
,
1994
).
27.
S. M.
Soskin
,
R.
Mannella
, and
O. M.
Yevtushenko
, “
Matching of separatrix map and resonant dynamics, with application to global chaos onset between separatrices
,”
Phys. Rev. E
77
,
036221
(
2008
).
28.
S. M.
Soskin
,
I. A.
Khovanov
, and
P. V. E.
McClintock
, “
Regular rather than chaotic origin of the resonant transport in superlattices
,”
Phys. Rev. Lett.
114
,
166802
(
2015
).
29.
W.
Kutta
, “
Beitrag zur näherungsweiser integration totaler differentialgleichungen
,”
Z. Math. Phys.
46
,
435
453
(
1901
).
30.
A.
Ralston
, “
Runge-Kutta methods with minimum error bounds
,”
Math. Comput.
16
,
431
437
(
1962
).
31.
Y. G.
Sinai
, “
Gibbs measure in ergodic theory
,”
Russ. Math. Surv.
27
,
21
64
(
1972
).
32.
V. S.
Anishchenko
,
I. A.
Khovanov
,
N. A.
Khovanova
,
D. G.
Luchinsky
, and
P. V. E.
McClintock
, “
Noise-induced escape from the Lorenz attractor
,”
Fluct. Noise Lett.
01
,
L27
L33
(
2001
).
33.
V. S.
Anishchenko
,
D. G.
Luchinsky
,
P. V. E.
McClintock
,
I. A.
Khovanov
, and
N. A.
Khovanova
, “
Fluctuational escape from a quasi-hyperbolic attractor in the Lorenz system
,”
J. Exp. Theor. Phys.
94
,
821
833
(
2002
).
34.
A.
Forrester
,
A.
Sobester
, and
A.
Keane
,
Engineering Design via Surrogate Modelling: A Practical Guide
(
John Wiley & Sons
,
Chichester
,
2008
).
35.
V. S.
Anishchenko
,
A. N.
Silchenko
, and
I. A.
Khovanov
, “
Synchronization of switching processes in coupled Lorenz systems
,”
Phys. Rev. E
57
,
316
322
(
1998
).
36.
I. A.
Khovanov
,
Data for stochastic approach for assessing the predictability of chaotic time series using reservoir computing
,
University of Warwick Publications service & WRAP. http://wrap.warwick.ac.uk/153069/
.
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