广东工业大学学报 ›› 2024, Vol. 41 ›› Issue (05): 111-118.doi: 10.12052/gdutxb.230121

• 微分方程及其应用 • 上一篇    下一篇

基于支持向量回归的边值问题求解方法

张慧菁, 莫艳   

  1. 广东工业大学 数学与统计学院, 广东 广州 510520
  • 收稿日期:2023-08-28 出版日期:2024-09-25 发布日期:2024-06-05
  • 通信作者: 莫艳(1986-),女,副教授,主要研究方向为计算及应用调和分析,E-mail:marvel-2008@163.com
  • 作者简介:张慧菁(1998-),女,硕士研究生,主要研究方向为计算及应用调和分析,E-mail:zhanghj9801@163.com
  • 基金资助:
    广州市科技计划项目(202102020704)

Support Vector Regression Based Method to Solve Boundary Value Problems

Zhang Hui-jing, Mo Yan   

  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2023-08-28 Online:2024-09-25 Published:2024-06-05

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摘要： 边值问题是方程问题中的一个热门领域，对常微分方程边值问题的研究已相当成熟。然而，当已知条件为离散点而非给定函数时如何求解边值问题仍有很大的研究空间。支持向量回归机是一种基于统计学习理论的机器学习方法，在逼近问题上有独特的优势，在经验风险最小化同时保证了泛化能力。因此，本文结合正则化、再生核理论和支持向量回归机对边值问题展开研究。将边值问题视为算子方程问题，利用再生核空间的性质得到方程解与已知条件的关系式，将问题转化为逼近问题后正则化为二次规划问题，然后利用支持向量回归机进行求解，最终得到一个由支持向量构成的稀疏解。通过Sobolev空间中的范数关系对所得数值解进行误差分析，给出了数值解与解析解的误差上界。以二阶三点边值问题为例，仅有离散值作为已知条件求解方程，实验结果表明该方法优于传统再生核方法和W-POAFD方法，验证了该方法的高精度和有效性。

关键词: 边值问题, 正则化, 再生核理论, 支持向量回归

Abstract: Boundary value problems are a hot research area in the field of equation problems, with the study of boundary value problems for ordinary differential equations being quite mature. However, there is significant research space in solving boundary value problems when the known conditions are discrete points rather than given functions. Support Vector Regression (SVR) is a machine learning method based on statistical learning theory, which shows unique advantages in approximation problems by minimizing empirical risk while ensuring generalization capability. Therefore, this paper combines regularization, reproducing kernel theory, and SVR to investigate boundary value problems. Treating the boundary value problem as an operator equation problem, the relationship between the solution of the equation and the known conditions is obtained using the properties of reproducing kernel spaces. The problem is then transformed into an approximation problem, regularized into a quadratic programming problem, and solved using SVR to obtain a sparse solution composed of support vectors. Error analysis of the numerical solution obtained is conducted using norms in Sobolev spaces, providing an upper bound for the error between the numerical solution and the analytical solution. Taking a second-order three-point boundary value problem as an example, where only discrete values are given as known conditions for solving the equation, experimental results demonstrate that this method outperforms traditional reproducing kernel methods and W-POAFD methods, confirming its high accuracy and effectiveness.

Key words: boundary value problem, regularization, reproducing kernel theory, support vector regression

中图分类号: 

  • O175.8
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