On Specific Features of an Approach Based on Feedforward Neural Networks to Solve Problems Based on Differential Equations

Abstract

To date, a multitude of methods have been developed for numerical solution of problems based on ordinary differential equations (ODEs) and partial differential equations (PDEs). The most common of these are finite-difference method, the finite-element method, and the finite-volume method. In this study, an alternative numerical approach is implemented, based on the approximation of functions by feedforward neural networks. The solution obtained using this approach is a differentiable analytical expression; in this respect it differs significantly from other methods that offer either discrete solutions or solutions with limited differentiability. In this study, we examine the influence of neural-network parameters (such as activation functions and weights in the error function) on the rate of convergence and accuracy of the obtained approximation of the solution for three types of differential equations: ordinary differential equations, integrable partial differential equations, and non-integrable partial differential equations. As model equations, we consider Korteweg–de Vries and Kudryashov–Sinelshchikov partial differential equations and second-order ordinary differential equations. In each case described above, the optimal ratios of the weight coefficients are found. The activation functions most efficient for each problem are determined.