An Orthogonal Polynomial Kernel-Based Machine Learning Model for Differential-Algebraic Equations

TL;DR

This paper introduces a novel machine learning approach based on orthogonal polynomial kernels, specifically Legendre polynomials, to solve complex differential-algebraic equations, expanding the application of LS-SVR methods with promising results.

Contribution

The study develops a new LS-SVR-based method for solving general DAEs using an operator approach linked to weighted residuals and orthogonal polynomials, demonstrating its effectiveness.

Findings

Successfully applied to nonlinear, fractional, integro-differential, and partial DAEs.

Outperforms existing state-of-the-art methods in accuracy and reliability.

Validated through extensive simulations across various DAE scenarios.

Abstract

The recent introduction of the Least-Squares Support Vector Regression (LS-SVR) algorithm for solving differential and integral equations has sparked interest. In this study, we expand the application of this algorithm to address systems of differential-algebraic equations (DAEs). Our work presents a novel approach to solving general DAEs in an operator format by establishing connections between the LS-SVR machine learning model, weighted residual methods, and Legendre orthogonal polynomials. To assess the effectiveness of our proposed method, we conduct simulations involving various DAE scenarios, such as nonlinear systems, fractional-order derivatives, integro-differential, and partial DAEs. Finally, we carry out comparisons between our proposed method and currently established state-of-the-art approaches, demonstrating its reliability and effectiveness.