Modeling nonlinear dynamics using an artificial neural network

Abstract

Technically an artificial neural network operates like a map: a
subspace of N-dimensional vector space is mapped onto a subspace of
M-dimensional space. Thus if M=N it can be considered as a sort of
discrete time dynamical system. Unlike usual systems where
nonlinearity are often introduced via power functions nonlinearity of
neural networks are typically sigmoid functions. A several theorems
are known that state that the neural networks can approximate with
arbitrary precision any nonlinear function. Due to this property they
are very promising object of analysis. In this work we construct and
discuss neural networks that model several famous ODE systems, namely
Lorenz, Chua and Roessler. Since training routines take into account
variations of control parameters, the resulting networks capture the
whole variety of regimes including the bifurcation transitions. The
reason why this is interesting is that the network structure remains
the same in all cases. Only its parameters are adjusted in different
ways during training to reproduced required dynamics. In the other
words we suggest a simple model map that captures withing one
universal class very different dynamical systems. Moreover this model
system can be implemented as a standalone physical devise using for
example PLDs (Programmable Logic Devices). Work of PVK on theoretical
formulation, elaboration of computer routines and numerical analysis
was supported by Russian Science Foundation, grant No 20-71-10048.

Speaker

Pavel Kuptsov
Laboratory of topological methods in dynamics, National Research University Higher School of Economics, Nizhny Novgorod, Russia
Russia

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