Meshfree Computational Algorithms Based on Normalized Radial Basis Functions

详细信息    查看全文

• 关键词：Modelling ; Boundary value problem (BVP) ; Identification problem ; Meshfree method ; Artificial neural network (ANN) ; Normalized radial basis function (NRBF) ; Training ; Error functional ; Optimization
• 刊名：Lecture Notes in Computer Science
• 出版年：2016
• 出版时间：2016
• 年：2016
• 卷：9719
• 期：1
• 页码：583-591
• 全文大小：346 KB
• 参考文献：1.Vasilyev, A.N., Tarkhov, D.A.: Neural Network Modelling Principles, Algorithms, Applications. SPbSPU Publishing House, Saint-Petersburg (2009). 528 (in Russian)
  2.Vasilyev, A.N., Osipov, V.P., Tarkhov, D.A.: Unified modelling process for physicotechnical objects with distributed constants. St. Petersburg State Polytechnical Univ. J. 3, 39–52 (2010). (in Russian)
  3.Bugmann, G.: Normalized radial basis function networks. Neurocomputing (Special Issue on Radial Basis Function Networks) 20, 97–110 (1998)
  4.Haykin, S.: Neural Networks: A Comprehensive Foundation, 2nd edn. Williams Publishing House, Jerusalem (2006). 1104 (in Russian)MATH
  5.Kolbin, I.S., Reviznikov, D.L.: Solving mathematical physics problems using normalized radial basis functions neural-like networks. Neurocomputers: Developing, Applications. Radiotekhnika vol. 2, pp. 12–19 (2012). (in Russian)
  6.Kolbin, I.S., Reviznikov, D.L.: Solving inverse mathematical physics problems using normalized radial basis functions neural-like networks. Neurocomputers: developing, applications. Radiotekhnika 9, 3–11 (2013). (in Russian)
  7.Panteleyev, A.V., Letova, T.A.: Optimization Techniques Through Examples and Tasks. Vysshaya shkola, Moscow (2005). 544 (in Russian)
  8.Hager, W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005)MathSciNet CrossRef MATH
  9.Chai, Z., Shi, B.: A novel lattice boltzmann model for the poisson equation. Appl. Math. Mech. 32, 2050–2058 (2008)MathSciNet MATH
  10.Kolbin, I.S.: Software package for the solution of mathematical modelling problems using neural networks techniques. Program Eng. 2, 25–30 (2013). (in Russian)
  11.Xie, O., Zhao, Z.: Identifying an unknown source in the Poisson equation by a modified Tikhonov regularization method. Int. J. Math. Comput. Sci. 6, 86–90 (2012)
  
• 作者单位：Alexander N. Vasilyev (16)
  Ilya S. Kolbin (17)
  Dmitry L. Reviznikov (18)
  
  16. Peter the Great St. Petersburg Polytechnical University, 29 Politechnicheskaya Street, 195251, Saint-petersburg, Russia
  17. Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, 44/2 Vavilov Street, 119333, Moscow, Russia
  18. Moscow Aviation Institute, 4 Volokolamskoye Shosse, 125993, Moscow, Russia
  
• 丛书名：Advances in Neural Networks ¨C ISNN 2016
• ISBN：978-3-319-40663-3
• 刊物类别：Computer Science
• 刊物主题：Artificial Intelligence and Robotics
  Computer Communication Networks
  Software Engineering
  Data Encryption
  Database Management
  Computation by Abstract Devices
  Algorithm Analysis and Problem Complexity
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1611-3349
• 卷排序：9719

文摘

In this paper, new methods for solving mathematical modelling problems, based on the usage of normalized radial basis functions, are introduced. Meshfree computational algorithms for solving classical and inverse problems of mathematical physics are developed. The distinctive feature of these algorithms is the usage of moving functional basis, which allows us to adapt to solution particularities and to maintain high accuracy at relatively low computational cost. Specifics of neural network algorithms application to non-stationary problems of mathematical physics were indicated. The paper studies the matters of application of developed algorithms to identification problems. Analysis of solution results for representative problems of source components (and boundary conditions) identification in heat transfer equations illustrates that the elaborated algorithms obtain regularization qualities and allow us to maintain high accuracy in problems with considerable measurement errors.