A Brief Bibliographic Review on Euler Neural Networks

Abstract

Euler Neural Networks are used exclusively for modeling non-linear dynamic systems, with application in control theory. These networks work coupled to the first order integrator of the Euler type. The Euler integrator was initially proposed by the mathematician Leonhard Paul Euler (1707-1783) in 1768. However, it is common knowledge that this type of numerical integrator is used with functions of instantaneous derivatives and its numerical precision is unsatisfactory. On the other hand, when this type of integrator is coupled with universal function approximators (e.g., artificial neural networks, Mamdani- Type fuzzy inference systems, etc.) its accuracy can be improved. The reason for this is that the artificial neural network, because it is a universal function approximator, it can learn the mean derivatives functions of the dynamic system considered, instead of instantaneous function derivatives. This small change makes the precision of the Euler integrator with mean derivatives equivalent to a Runge-Kutta of any order. The only drawback of the Euler Integrator, designed with mean derivatives is that it has a fixed step, while the Euler integrator designed with instantaneous derivatives is of varying step. In the literature, when coupled with a numerical integrator with a universal function approximator, such structures are known as universal numerical integrators. This article proposes to make a rather brief description of Euler Neural Networks coupled with a literature revision, to spread its use in the scientific community, since this structure of neural model structure is little known.