Orthogonal Embedding-Based Artificial Neural Network Solutions to Ordinary Differential Equations

Providing numerical solutions to differential equations in cases where analytical solutions are not available is of great importance. Recently, obtaining more accurate numerical solutions with artificial neural network-based machine learning methods are seen as promising developments for numerical solutions of differential equations. In this paper, a low-cost, orthogonal embedding-based network with fast training by simple gradient descent algorithm is proposed to obtain numerical solutions of differential equations. This architecture is essentially a two-layer neural network that takes orthogonal polynomials as input. The efficiency and accuracy of the method used in this paper are demonstrated in various problems and comparisons are made with other methods. It is observed that the proposed method stands out especially when compared with high-cost solutions.