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.

翻译：最近引入的用于求解微分和积分方程的最小二乘支持向量回归（LS-SVR）算法引起了广泛关注。在本研究中，我们将该算法的应用扩展到求解微分代数方程（DAE）系统。我们的工作提出了一种新颖的方法，通过建立LS-SVR机器学习模型、加权残差法与勒让德正交多项式之间的联系，以算子形式求解一般DAE。为评估所提方法的有效性，我们针对各种DAE场景进行了仿真，涵盖非线性系统、分数阶导数、积分微分方程以及偏微分代数方程。最后，我们将所提方法与当前最先进的现有方法进行比较，证明了其可靠性和有效性。