A Qualitative Approach to Universal Numerical Integrators (UNIs) with Computational Application

Abstract Universal Numerical Integrators (UNIs) can be defined as the coupling of a universal approximator of functions (e.g., artificial neural network) with some conventional numerical integrator (e.g., Euler or Runge–Kutta). The UNIs are used to model non-linear dynamic systems governed by Ordinary Differential Equations (ODEs). Among the main types of UNIs existing in the literature, we can mention (i) The Euler-Type Universal Numerical Integrator (E-TUNI), (ii) The Runge-Kutta Neural Network (RKNN), and (iii) The Non-linear Auto Regressive Moving Average with Exogenous input or NARMAX model. All of them are equally accurate, regardless of their order. Furthermore, one of the reasons for writing this article is to show the reader that there are many other UNIs besides these. Thus, this article aims to carry out a detailed bibliographic review of this object of study, taking into more significant consideration the qualitative aspects of these UNIs. Computational experiments are also presented in this article to prove the numerical effectiveness of the main types of UNIs in the literature. Therefore, it is expected that this paper will help researchers in the future development of new UNIs.