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 Rus. J. Nonlin. Dyn., 2021, Volume 17, Number 1, Pages 5–21 (Mi nd738)

Nonlinear physics and mechanics

Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems

P. V. Kuptsova, A. V. Kuptsovab, N. V. Stankevicha

a Laboratory of topological methods in dynamics, National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, Nizhny Novgorod, 603155 Russia
b Institute of electronics and mechanical engineering, Yuri Gagarin State Technical University of Saratov, ul. Politekhnicheskaya 77, Saratov, 410054 Russia

Abstract: We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. The theoretical benefit from this approach is that the universal model admits applying common mathematical methods without needing to develop a unique theory for each particular dynamical equations. From the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Rössler system and also the Hindmarch–Rose model. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. A high similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents.

Keywords: neural network, dynamical system, numerical solution, universal approximation theorem, Lyapunov exponents

 Funding Agency Grant Number Russian Science Foundation 20-71-10048 The work of PVK on theoretical formulation and numerical computations and the work of NVS on results analysis were supported by a grant of the Russian Science Foundation No. 20-71-10048.

DOI: https://doi.org/10.20537/nd210102

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MSC: 65P20, 37M05, 65L05
Accepted:15.03.2021
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Citation: P. V. Kuptsov, A. V. Kuptsova, N. V. Stankevich, “Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems”, Rus. J. Nonlin. Dyn., 17:1 (2021), 5–21

Citation in format AMSBIB
\Bibitem{KupKupSta21} \by P. V. Kuptsov, A. V. Kuptsova, N. V. Stankevich \paper Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems \jour Rus. J. Nonlin. Dyn. \yr 2021 \vol 17 \issue 1 \pages 5--21 \mathnet{http://mi.mathnet.ru/nd738} \crossref{https://doi.org/10.20537/nd210102} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4240814} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85105223065}