Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Generative Adversarial Network for Detecting Cyber Threats in Industrial Systems Read chapter Read first chapter

2021 | OriginalPaper | Chapter

Solution of Partial Differential Equations on Radial Basis Functions Networks

Authors : Mohie Alqezweeni, Vladimir Gorbachenko

Published in: Proceedings of International Scientific Conference on Telecommunications, Computing and Control

Publisher: Springer Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The solution of boundary value problems described by partial differential equations on networks of radial basis functions is considered. An analysis of gradient learning algorithms for radial basis functions networks showed that the widely used first-order method, the gradient descent method, does not provide a high learning speed and solution accuracy. The fastest method of the second order, the trust region method, is very complex. A learning algorithm based on the Levenberg–Marquardt method is proposed. The proposed algorithm, with a simpler implementation, showed comparable results in comparison with the trust region method.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Computer Model for Analysis of the Process of Image Construction in Optical-Electronic Visualization Systems
next chapter Predicting Personality from Image Preferences: Tendencies, Models and Implementation
Literature
1.
Grieves, M.: Digital twin: manufacturing excellence through virtual factory replication. Nc-Race 18 95, 6–15 (2014)
2.
Madni, A., Madni, C., Lucero, S.: Leveraging digital twin technology in model-based systems engineering. Systems 7(1), 7 (2019)CrossRef
3.
Farlow, S.J.: Partial Differential Equations for Scientists and Engineers. Dover Publications, USA (1993)
4.
Mazumder, S.: Numerical methods for partial differential equations: finite difference and finite volume methods. In: Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods, pp. 1–461. Elsevier, Amsterdam (2015)
5.
Tolstykh, A.I., Shirobokov, D.A.: Meshless method based on radial basis functions. Comput. Math. Math. Phys. 45(8), 1447–1454 (2005)MathSciNetMATH
6.
Vasilyev, A., Tarkhov, D., Malykhina, G.: Methods of creating digital twins based on neural network modeling. Mod. Inf. Technol. IT-Educ. 14(3), 521–532 (2018)
7.
Griebel, M., Schweitzer, M.A.: Meshfree Methods for Partial Differential Equations III. Lecture Notes in Computational Science and Engineering (2007)
8.
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2004)
9.
Chen, W., Fu, Z.-J.: Recent Advances in Radial Basis Function Collocation Methods. Springer, Berlin (2014)
10.
Yadav, N., Yadav, A., Kumar, M.: An Introduction to Neural Network Methods for Differential Equations. Springer, Berlin (2015)
11.
Aggarwal, C.C.: Neural Networks and Deep Learning. Springer International Publishing, Berlin (2018)
12.
Jianyu, L., Siwei, L., Yingjian, Q., Yaping, H.: Numerical solution of elliptic partial differential equation by growing radial basis function neural networks. Neural Netw. 16(5–6), 729–734 (2003)CrossRef
13.
Mai-Duy, N.: Solving high order ordinary differential equations with radial basis function networks. Int. J. Numer. Meth. Eng. 62(6), 824–852 (2005)MathSciNetCrossRef
14.
Vasiliev, A.N., Tarkhov, D.A.: Neural Network Modeling: Principles. Algorithms. Applications. St. Petersburg Polytechnic University Publishing House (2009)
15.
Gorbachenko, V.I., Zhukov, M.V.: Solving boundary value problems of mathematical physics using radial basis function networks. Comput. Math. Math. Phys. 57(1), 145–155 (2017)MathSciNetCrossRef
16.
Gorbachenko, V.I., Lazovskaya, T.V., Tarkhov, D.A., Vasilyev, A.N., Zhukov, M.V.: Neural network technique in some inverse problems of mathematical physics. In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9719, pp. 310–316. Springer, Berlin (2016)
17.
Fasshauer, G.E., Zhang, J.G.: On choosing “optimal” shape parameters for RBF approximation. Numer. Algorithms 45(1–4), 345–368 (2007)MathSciNetCrossRef
18.
Kansa, E.J.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates. Comput. Math. Appl. 19(8–9), 127–145 (1990)MathSciNetCrossRef
19.
Kansa, E.J.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8–9), 147–161 (1990)MathSciNetCrossRef
20.
Niyogi, P., Girosi, F.: On the relationship between generalization error, hypothesis complexity, and sample complexity for radial basis functions. Neural Comput. 8(4), 819–842 (1996)CrossRef
21.
Watkins, D.S.: Fundamentals of Matrix Computations. Wiley, Hoboken (2005)
22.
Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016)
23.
Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Emerald Group Publishing, UK (1982)
24.
Jia, W., Zhao, D., Shen, T., Su, C., Hu, C., Zhao, Y.: A new optimized GA-RBF neural network algorithm. Comput. Intell. Neurosci. 2014, 1–6, Article ID 982045 (2014)
25.
26.
Polak, E., Ribiere, G.: Note sur la convergence de méthodes de directions conjuguées. Revue Française d’informatique et de Recherche Opérationnelle. Série Rouge 3(16), 35–43 (1969)
27.
Nocedal, J., Wright, S.: Numerical Optimization. Springer, Berlin (2006)
28.
Zhang, L., Li, K., Wang, S.: An improved conjugate gradient algorithm for radial basis function (RBF) networks modelling. In: Proceedings of the 2012 UKACC International Conference on Control, CONTROL 2012, pp. 19–23 (2012)
29.
Gorbachenko, V.I., Artyukhina, E.V.: Mesh-free methods and their implementation with radial basis neural networks. Neirokomp’yutory: Razrabotka, Primentnine 11, 4–10 (2010) (in Russian)
30.
Alqezweeni, M.M., Gorbachenko, V I., Zhukov, M.V., Jaafar, M.S.: Efficient solving of boundary value problems using radial basis function networks learned by trust region method. Int. J. Math. Math. Sci. 2018, 1–4, Article ID 9457578 (2018).
31.
Elisov, L.N., Gorbachenko, V.I., Zhukov, M.V.: Learning radial basis function networks with the trust region method for boundary problems. Autom. Remote Control 79(9), 1621–1629 (2018)MathSciNetCrossRef
32.
Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust Region Methods. Trust Region Methods. Society for Industrial and Applied Mathematics, USA (2000)
33.
Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963)MathSciNetCrossRef
34.
Boyd, J.P., Gildersleeve, K.W.: Numerical experiments on the condition number of the interpolation matrices for radial basis functions. Appl. Numer. Math. 61(4), 443–459 (2011)MathSciNetCrossRef
35.
Sutskever, I., Martens, J., Dahl, G., Hinton, G.: On the importance of initialization and momentum in deep learning. In: 30th International Conference on Machine Learning, ICML 2013, pp. 2176–2184. International Machine Learning Society (IMLS) (2013)
Metadata
Title
Solution of Partial Differential Equations on Radial Basis Functions Networks
Authors
Mohie Alqezweeni
Vladimir Gorbachenko
Copyright Year
2021
Publisher
Springer Singapore
Book
Proceedings of International Scientific Conference on Telecommunications, Computing and Control

Print ISBN: 978-981-336-631-2

Electronic ISBN: 978-981-336-632-9

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-981-33-6632-9_42