Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems

Russian Journal of Nonlinear Dynamics. 2021. Vol. 17. No. 1. P. 5–21.

Kuptsov P., Kuptsova A. V., Stankevich N.

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 Rцssler 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: нейронная сеть, динамическая система, показатели Ляпунова, Lyapunov exponents, dynamical system, Neural Network, universal approximation theorem