Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real-world applications. This work is devoted to investigating the effective dynamics for slow–fast stochastic dynamical systems. Given observation data on a short-term period satisfying some unknown slow–fast stochastic systems, we propose a novel algorithm, including a neural network called Auto-SDE, to learn an invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also validated to be accurate, stable, and effective through numerical experiments under various evaluation metrics.
REFERENCES
1.
A. C.
Fowler
and G. C.
Kember
, “The Lorenz-Krishnamurthy slow manifold
,” J. Atmos. Sci.
53
, 1433
–1437
(1996
). 2.
D. A.
Goussis
, “The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation-oscillations
,” Physica D
248
, 16
–32
(2013
). 3.
B. K.
Carpenter
, G. S.
Ezra
, S. C.
Farantos
, Z. C.
Kramer
, and S.
Wiggins
, “Empirical classification of trajectory data: An opportunity for the use of machine learning in molecular dynamics
,” J. Phys. Chem. B
122
(13
), 3230
–3241
(2018
). 4.
G.
Zhang
, Y.
Zeng
, P.
Gordiichuk
, and M. S.
Strano
, “Chemical kinetic mechanisms and scaling of two-dimensional polymers via irreversible solution-phase reactions
,” J. Chem. Phys.
154
(19
), 194901
(2021
). 5.
6.
7.
8.
9.
N.
Berglund
and B.
Gentz
, Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach
(Springer Science & Business Media
, 2006
).10.
G.
Pavliotis
and A.
Stuart
, Multiscale Methods: Averaging and Homogenization
(Springer Science & Business Media
, 2008
).11.
J.
Duan
, An Introduction to Stochastic Dynamics
(Cambridge University Press
, 2015
), Vol. 51.12.
D. L.
Donoho
, “High-dimensional data analysis: The curses and blessings of dimensionality,” in AMS Lectures on Math Challenges (AMS, 2000), Vol. 1, p. 32.13.
B.
Krauskopf
, H. M.
Osinga
, E. J.
Doedel
, M. E.
Henderson
, J. M.
Guckenheimer
, A.
Vladimirsky
, M.
Dellnitz
, and O.
Junge
, “A survey of methods for computing (un)stable manifolds of vector fields
,” Int. J. Bifurcation Chaos
15
, 763
–791
(2005
). 14.
M.
Branicki
and S.
Wiggins
, “An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems
,” Physica D
238
, 1625
–1657
(2009
). 15.
S.
Jain
and G.
Haller
, “How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models
,” Nonlinear Dyn.
107
(2
), 1417
–1450
(2022
). 16.
B.
Haasdonk
, B.
Hamzi
, G.
Santin
, and D.
Wittwar
, “Kernel methods for center manifold approximation and a weak data-based version of the center manifold theorem
,” Physica D
427
, 133007
(2021
). 17.
C. W.
Gear
, T. J.
Kaper
, I. G.
Kevrekidis
, and A.
Zagaris
, “Projecting to a slow manifold: Singularly perturbed systems and legacy codes
,” SIAM J. Appl. Dyn. Syst.
4
, 711
–732
(2005
). 18.
J.
Duan
, K.
Lu
, and B.
Schmalfuss
, “Invariant manifolds for stochastic partial differential equations
,” Ann. Probab.
31
, 2109
–2135
(2003
). 19.
B.
Schmalfuß
and K.
Schneider
, “Invariant manifolds for random dynamical systems with slow and fast variables
,” J. Dyn. Differ. Equ.
20
, 133
–164
(2008
). 20.
J.
Ren
, J.
Duan
, and C.
Jones
, “Approximation of random slow manifolds and settling of inertial particles under uncertainty
,” J. Dyn. Differ. Equ.
27
, 961
–979
(2015
). 21.
H.
Fu
, X.
Liu
, and J.
Duan
, “Slow manifolds for multi-time-scale stochastic evolutionary systems
,” Commun. Math. Sci.
11
(1
), 141–162
(2013
).22.
R. Z.
Khasminskij
, “On the principle of averaging the Itov’s stochastic differential equations
,” Kybernetika
4
(3
), 260
–279
(1968
); available at https://www.kybernetika.cz/content/1968/3/260.23.
N.
Namachchivaya
, R.
Sowers
, and L.
Vedula
, “Stochastic averaging on graphs: Noisy Duffing-van der Pol equation
,” AIP Conf. Proc.
502
, 255
–265
(2000
).24.
W.
Wang
and A. J.
Roberts
, “Slow manifold and averaging for slow–fast stochastic differential system
,” J. Math. Anal. Appl.
398
, 822
–839
(2013
). 25.
W.
Liu
, M.
Röckner
, X.
Sun
, and Y.
Xie
, “Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients
,” J. Differ. Equ.
268
(6
), 2910
–2948
(2020
). 26.
C. J.
Dsilva
, R.
Talmon
, C. W.
Gear
, R. R.
Coifman
, and I. G.
Kevrekidis
, “Data-driven reduction for a class of multiscale fast-slow stochastic dynamical systems
,” SIAM J. Appl. Dyn. Syst.
15
, 1327
–1351
(2016
). 27.
F. X.-F.
Ye
, S.
Yang
, and M.
Maggioni
, “Nonlinear model reduction for slow-fast stochastic systems near manifolds,” arXiv:2104.02120 (2021).28.
P. R.
Vlachas
, G.
Arampatzis
, C.
Uhler
, and P.
Koumoutsakos
, “Multiscale simulations of complex systems by learning their effective dynamics
,” Nat. Mach. Intell.
4
, 359
–366
(2022
). 29.
V.
Krajňák
, S.
Naik
, and S.
Wiggins
, “Predicting trajectory behaviour via machine-learned invariant manifolds
,” Chem. Phys. Lett.
789
, 139290
(2022
).30.
L.
Manoni
, C.
Turchetti
, and L.
Falaschetti
, “An effective manifold learning approach to parametrize data for generative modeling of biosignals
,” IEEE Access
8
, 207112
–207133
(2020
). 31.
F.
Lu
, K. K.
Lin
, and A. J.
Chorin
, “Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation
,” Physica D
340
, 46
–57
(2017
). 32.
A. J.
Linot
and M. D.
Graham
, “Deep learning to discover and predict dynamics on an inertial manifold
,” Phys. Rev. E
101
(6
), 062209
(2020
). 33.
W.
Liao
, M.
Maggioni
, and S.
Vigogna
, “Multiscale regression on unknown manifolds,” arXiv:2101.05119 (2021).34.
J. T.
Vogelstein
, E. W.
Bridgeford
, M.
Tang
, D.
Zheng
, C. B.
Douville
, R. C.
Burns
, and M.
Maggioni
, “Supervised dimensionality reduction for big data
,” Nat. Commun.
12
, 2872
(2021
). 35.
A. V.
Bernstein
and A. P.
Kuleshov
, “Manifold learning: Generalization ability and tangent proximity
,” Int. J. Softw. Inf.
7
(3
), 359
–390
(2013
); available at http://www.ijsi.org/ijsi/article/abstract/i165.36.
N.
Berglund
and B.
Gentz
, “Geometric singular perturbation theory for stochastic differential equations
,” J. Differ. Equ.
191
, 1
–54
(2003
). 37.
Z.
Schuss
, Theory and Applications of Stochastic Processes: An Analytical Approach
(Springer Science & Business Media
, 2009
), Vol. 170.38.
G. E.
Hinton
and R.
Salakhutdinov
, “Reducing the dimensionality of data with neural networks
,” Science
313
, 504
–507
(2006
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
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