Abstract

Abstract In this work we explore the use of deep learning models based on deep feedforward neural networks to solve ordinary and partial differential equations. The illustration of this methodology is given by solving a variety of initial and boundary value problems. The numerical results, obtained based on different feedforward neural networks structures, activation functions and minimization methods, were compared to each other and to the exact solutions. The neural network was implemented using the Python language, with the Tensorflow library.

Highlights

  • Differential equations play an important role in various fields of science and technology, modeling real-world problems of great interest in society

  • Classical numerical methods are based on the discretization of the domain at a finite number of points or elements, so that the approximate solution is defined at these points

  • In the present work we explore the use of feedforward neural networks in the solution of boundary and initial value problems, involving ordinary and partial differential equations

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Summary

Introduction

Differential equations play an important role in various fields of science and technology, modeling real-world problems of great interest in society. In [15,19] the authors highlighted some properties of the NN solution in comparison with conventional numerical methods, such as: the solution is continuous and differentiable over all the domain; the propagation of rounding errors, common in classic numerical methods, does not affect the solution; the computational complexity does not increase considerably with the number of discretization points and with the number of dimensions; the method can be applied in the solution of linear, nonlinear, steady or unsteady ordinary or partial equations Another motivation in using neural networks to solve differential equations is its property of approximating continuous functions. In the present work we explore the use of feedforward neural networks in the solution of boundary and initial value problems, involving ordinary and partial differential equations.

Feedforward Neural Networks
Description of the Method
Numerical Experiments
Ordinary Differential Equation (ODE)
The Poisson Problem
Heat Equation
Conclusions
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