Modified least squares method and a review of its applications in machine learning and fractional differential/integral equations

TL;DR

This paper introduces a modified least squares method based on fractional orthogonal polynomials, demonstrating its advantages over classical methods and exploring its applications in fractional differential equations and machine learning.

Contribution

The paper presents a novel modified least squares method utilizing fractional orthogonal polynomials and discusses its applications in fractional differential equations and machine learning.

Findings

The modified least squares method outperforms the classical approach in numerical experiments.

The method effectively solves problems in fractional differential and integral equations.

Applications in machine learning show promising results.

Abstract

The least squares method provides the best-fit curve by minimizing the total squares error. In this work, we provide the modified least squares method based on the fractional orthogonal polynomials that belong to the space $M_{n}^{\lambda} := \text{span}\{1,x^{\lambda},x^{2\lambda},\ldots,x^{n\lambda}\},~\lambda \in (0,2]$. Numerical experiments demonstrate how to solve different problems using the modified least squares method. Moreover, the results show the advantage of the modified least squares method compared to the classical least squares method. Furthermore, we discuss the various applications of the modified least squares method in the fields like fractional differential/integral equations and machine learning.