Algorithms for Solving Ordinary Differential Equations Based on Orthogonal Polynomial Neural Networks

The paper proposes single-layer neural network algorithms for solving ordinary differential equations of the second order, built on the principles of functional link. According to this principle, the hidden layer of the neural network is replaced by a functional expansion block to improve input patterns using orthogonal Chebyshev, Legendre and Laguerre polynomials. The algorithms of polynomial neural networks were implemented in the Python programming language in the PyCharm environment. The operation of the algorithms of polynomial neural networks was tested by solving the initial and boundary value problems for the nonlinear Lane-Emden equation. The results of the solution are compared with the exact solution of the problems under consideration, as well as with the solution obtained using a multilayer perceptron. It is shown that polynomial neural networks can work more efficiently than multilayer neural networks. The issues of overfiting of polynomial neural networks and scenarios for overcoming it are considered.