Artificial neural network as a universal model of nonlinear dynamical systems

TL;DR

This paper introduces a neural network-based universal map that can replicate the behavior of various nonlinear dynamical systems directly from their ODEs, capturing bifurcations and enabling new analytical and numerical approaches.

Contribution

It presents a neural network model that encodes dynamical systems from their ODEs without numerical simulation, capturing bifurcations and enabling universal analysis.

Findings

High similarity between neural network model and numerical solutions for Lorenz, Rössler, and Hindmarch-Rose systems.

The model accurately reproduces attractors, spectra, bifurcation diagrams, and Lyapunov exponents.

The approach offers a new analytical and computational tool for studying nonlinear dynamical systems.

Abstract

We suggest a universal map capable to recover a 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. Theoretical benefit from this approach is that the universal model admits using common mathematical methods without needing to develop a unique theory for each particular dynamical equations. Form 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 Roessler system and also Hindmarch-Rose neuron. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. High similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents.