Investigation of the Functional Stability of Neural Network Algorithm for Solving the Ordinary Differential Equations

Abstract

The paper analyzes the neural network approach for solving the Cauchy problem of ordinary differential equations of the first order, based on the representation of the function as a superposition of elementary functions. The use of neural network approach allows obtaining the desired solution in the form of a functional dependence, which satisfies the required conditions of smoothness. On the basis of a two-layer perceptron, a model of neural network solution of the problem and a numerical algorithm implementing the search for a solution are constructed. The software-algorithmic solution of the Cauchy problem is obtained. To determine the stability of the neural network approach, a series of experiments were conducted to find a solution to a particular Cauchy problem of ordinary differential equation of the first order with an analytical solution. The study shows that the considered neural network algorithm has no functional stability. This may be due to the problems of weights minimization, scalability in network training and other factors. Keywords—ordinary differential equations, artificial neural network, optimization methods, functional stability