Least squares support vector regression for differential equations on unbounded domains. (October 2021)

• Record Type: Journal Article
• Authors: Pakniyat, A.; Parand, K.; Jani, M.
• Is Part Of: Chaos, solitons and fractals. Volume 151(2021)
• Publication Date: 2021-10
• Subjects: Least squares support vector regression -- Unbounded domain -- Fractional differential equations -- Hermite kernel -- Galerkin LS-SVR -- Collocation LS-SVR; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2021.111232
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: A machine learning algorithm based on least squares support vector machines for function estimation with two different approaches using the orthogonal Hermite kernel is proposed for the numerical simulation of ordinary and differential equations. The training points are taken as the roots of the Hermite polynomials that are used as collocation points in the kernel of least square support vector machines for regression. The formulation of the method in both approaches is discussed in detail and the sparsity of the resulting linear systems as well as the computational complexity is illustrated through some test examples. The method is supported with some numerical examples with different behaviors at infinity as well as a presentation of the structure of the involving matrices. Abstract: In this paper, a numerical method based on the least-squares support vector regression, and spectral methods are developed for solving differential equations on unbounded domains. In the proposed method, Hermite functions are used as the orthogonal kernel of the support vector regression. The resulting optimization problem is then reduced to a linear system in both collocation and Galerkin approaches of the method. The systems are then analyzed, along with a discussion of the sparsity of the involving matrices. Providing some numerical examples, including fractional differential equations, the accuracy and efficiency of the method are illustrated and compared with some existing methods.