On the Equivalence of Real Dynamic Process and Its Neural Network Quadratic Models

Vasiliy Ye Belozyorov,Yevhen V Koshel,Danylo V Dantsev

doi:10.15421/142109

Abstract

A dynamic process defined by its own time series is considered. Using the methods of qualitative recurrent analysis, the dimension of the embedding space and the optimal time delay of the specified series are determined. Using these characteristics, a neural network with a quadratic activation function is modeled. The simulation result is presented in the form of a system of neural ODEs. After that, the Lyapunov exponents of the real dynamic system and its neural network model are calculated. Then the closeness of these exponents for a real system and its model makes it possible to judge the adequacy (equivalence) of both dynamic processes. Examples are given.

Highlights

One of the fundamental goals of machine learning is modeling and understanding real-world phenomena from observations
Time series (1.1) will describe any dynamic process obtained from the observation of any one variable characterizing this process, which is measured at regular intervals
The article proposes a new algorithm for reconstructing a system of quadratic differential equations from a known time series

Summary

Introduction

One of the fundamental goals of machine learning is modeling and understanding real-world phenomena from observations. Be a system of ordinary autonomous differential equations and let x(t, x0) be a trajectory of this system with initial data x0 ∈ Rn. Here F(x) : Rn → Rn is a continuous vector-function; x(0, x0) = x0. In the case of nonlinear systems, the necessary equivalence condition is the coincidence of the Lyapunov exponents for systems (1.2) and (1.3). Experiments show that the most logical approach to describing processes in mechanical, hydrodynamic and electrical models is based on the use of well-known physical laws. These can be the laws of conservation of energy.

Generalization to n-dimensional case

Put Y T

Examples

Conclusion