广东工业大学学报 ›› 2020, Vol. 37 ›› Issue (02): 87-93.doi: 10.12052/gdutxb.190050

• 综合研究 • 上一篇    下一篇

基于支持向量机的离散线性微分方程求解方法

王丹荣, 莫艳   

  1. 广东工业大学 应用数学学院, 广东 广州 510520
  • 收稿日期:2019-04-04 出版日期:2020-03-10 发布日期:2020-02-12
  • 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 E-mail:marvel-2008@163.com
  • 作者简介:王丹荣(1991-),女,硕士研究生,主要研究方向为计算及应用调和分析
  • 基金资助:
    国家自然科学基金资助项目（11801095，11626067）；广东省自然科学基金资助项目（2017A030310538）；广东省高校特色创新项目（2017KTSCX062）；广东工业大学青年百人科研启动基金资助项目（220413131，220413550）

Support Vector Machines Based Method to Solve Discrete Linear Differential Equations

Wang Dan-rong, Mo Yan   

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2019-04-04 Online:2020-03-10 Published:2020-02-12

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摘要： 支持向量机（Support Vector Machines，SVMs）在逼近问题的求解上展现出了良好的有效性和可行性，而微分方程求解问题是许多学者研究的热门课题。其中，离散微分方程及其逆问题的求解具有十分重要的意义。本文将支持向量机、Tikhonov正则化和再生核理论相结合，提出一种求解离散线性微分方程及其逆问题的方法。该方法适用于一般的离散线性微分方程及其逆问题的求解，能够得到具有解析表达式的稀疏解，便于后续应用。实验表明，所提出的方法是有效的。

关键词: 离散线性微分方程, 逆问题, Tikhonov正则化, 再生核, 支持向量机

Abstract: Support vector machines (SVMs) show excellent effectiveness and feasibility in solving approximation problems. Solving differential equations is a hot topic studied by many scholars, in which the solutions of discrete differential equations and their inverse problems are of great significance. A method of solving discrete linear differential equations and their inverse problems is proposed by combining support vector machines, Tikhonov regularization and the theory of reproducing kernels. This method is suitable for solving general discrete linear differential equations and their inverse problems. This method can obtain sparse solutions with analytical expressions, which is convenient for subsequent applications. Experiments show that the proposed method is effective.

Key words: discrete linear differential equations, inverse problems, Tikhonov regularization, reproducing kernel, support vector machines

中图分类号: 

  • O172.1
[1] LAGARIS I E, LIKAS A, FOTIADIS D I. Artificial neural networks for solving ordinary and partial differential equations[J]. IEEE Transactions on Neural Networks and Learning Systems, 1998, 9(5):987-1000
[2] MEADE A J J, FERNANDEZ A A. The numerical solution of linear ordinary differential equations by feedforward neural networks[J]. Mathematical & Computer Modelling Math, 1994, 19(12):1-25
[3] VAPNIK V. The Nature of Statistical Learning Theory[M]. New York:Springer-Verlag, 1995.
[4] AHMED S, KHALID M, AKRAM U. A method for short-term wind speed time series forecasting using support vector machine regression model[C]//20176th International Conference on Clean Electrical Power. Santa Margherita Ligure:IEEE, 2017:190-195.
[5] WANG X D. Forecasting short-term wind speed using support vector machine with particle swarm optimization[C]//2017 International Conference on Sensing, Diagnostics, Prognostics and Control. Shanghai:IEEE, 2017:241-245.
[6] PAN J, YANG B, CAI S B, et al. Finger motion pattern recognition based on sEMG support vector machine[C]//2017 IEEE International Conference on Cyborg & Bionic Systems. Beijing:IEEE, 2017.
[7] WU W, ZHOU H. Data-Driven diagnosis of cervical cancer with support vector machine-based approaches[J]. IEEE Access, 2017, 5:25189-25195
[8] POLAT H, DANAEI M H, CETIN A. Diagnosis of chronic kidney disease based on support vector machine by feature selection methods[J]. Journal of Medical Systems, 2017, 41(4):55
[9] TAIE S A, GHONAIM W. Title CSO-based algorithm with support vector machine for brain tumor's disease diagnosis[C]//2017 IEEE International Conference on Pervasive Computing & Communications Workshops. Kona:IEEE, 2017.
[10] MO Y, QIAN T. Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem[J]. Applied Mathematics and Computation, 2014, 245:509-519
[11] SUYKENS J A K, VANDEWALLE J. Least squares support vector machine classifiers[J]. Neural Processing Letters, 1999, 9(3):293-300
[12] MEHRKANOON S, FALCK T, SUYKENS J A K. Approximate solutions to ordinary differential equations using least squares support vector machines[J]. IEEE Transactions on Neural Networks and Learning Systems, 2012, 23(9):1356-1367
[13] ZHOU S S, WANG B J, CHEN L. High precision approximate analytical solutions to ODE using LS-SVM[J]. The Journal of China Universities of Posts and Telecommunications, 2018, 25(4):94-102
[14] CASTRO L P, SAITOH S, SAWANO Y, et al. Discrete linear differential equations[J]. International Mathematical Journal of Analysis & Its Applications, 2012, 32(3):181-191
[15] MATSUURA T, SAITOH S, TRONG D D. Numerical solutions of the Poisson equation[J]. Applicable Analysis, 2004, 83(10):1037-1051
[16] SAITOH S. Integral Transforms, Reproducing Kernels and their applications[J]. Journal of Experimental Medicine, 1997, 188(1):39-48
[17] SAITOH S. Approximate real inversion formulas of the gaussian convolution[J]. Applicable Analysis, 2004, 83(7):727-733
[18] 邓乃扬, 田英杰. 支持向量机:理论、算法与拓展[M]. 北京:科学出版社, 2009.
[19] BOYD S, VANDENBERGHE L. Convex Optimization[M]. Cambridge University Press, 2004.
[20] RIEGER C, ZWICKNAGL B. Deterministic error analysis of support vector regression and related regularized kernel methods[J]. Journal of Machine Learning Research, 2006, 10(5):2115-2132
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