Bridging machine learning and weighted residual methods for delay differential equations of fractional order

Abstract

In this research, an attempt has been made to build a bridge between a machine learning model called Least-Squares Support Vector Regression (LS-SVR) and Weighted Residual Methods (WRMs) to solve delay differential equations in fractional derivative order. Our machine learning algorithm, which is influenced by the collocation method, uses Legendre polynomials as a kernel function to approximate the solution of linear and nonlinear delay differential equations, including the Lane–Emden pantograph equation and some fractional cases. In the proposed algorithm, the estimated solution of delay differential equations is approximated as a linear combination of M degrees of Legendre polynomials and a set of weights that are learned during the fitting process. The roots of the Legendre functions are utilized as training data to develop the algorithm. The effectiveness of our method is demonstrated through the use of numerical graphs and tables. A comparative analysis of the numerical results has shown that the proposed method yields better accuracy and convergence compared to other numerical techniques.