EULER'S METHOD AND NEURAL NETWORKS FOR THE NUMERICAL APPROXIMATION OF ORDINARY DIFFERENTIAL EQUATIONS

Abstract

In this work, we propose the use of numerical methods to solve differential equations and the
use of machine learning techniques to obtain approximate solutions of differential equations.
More specifically, the present work aims at the implementation of Euler's Method together with
neural network models for the approximation of solutions of ordinary differential equations. As
a methodology, we apply Euler's method to obtain approximate values of the solution of the
differential equation near the initial point. We then use these points to train a neural network
model as an approximation of the real solution over an extended interval. The final model is
close to the neural network trained with points of the exact solution of the differential equation
and shows to be better when compared to the direct application of the Euler method in the entire
interval under consideration. In particular, the analysis shows that the proposed model is
indicated in cases where Euler's method lacks stability.