Soluções de equações diferenciais via redes neurais artificiais

Abstract

The study of differential equations plays an important role in several fields of science and technology, through the modeling of real-world problems. As most of the mathematical models described by differential equations (ordinary and partial) do not have an analytical solution, numerical methods, such as finite differences and finite elements, are widely used to solve it. Recently, many studies have been dedicated to the application of deep artificial neural networks, known as deep learning, in the solution of differential equations, with promising results. The aim of this work is to explore the use of artificial neural networks feedforward in the solution of ordinary and partial differential equations. The neural network was implemented using the Python language, with the Tensorflow library. We applied this methodology in the solution of two initial value problems, in the Poisson problem (two-dimensional), in two unsteady problems (heat and wave equations) and in a singularly perturbed one-dimensional problem (convection-diffusion equation) to evaluate the quality of the solutions obtained . Some comparisons with classical numerical methods, such as Euler, Runge-Kutta, finite differences and finite elements, are presented.