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Infinite Order Systems of Differential Equations and Large Scale Random Neural Networks

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Analytical and Computational Methods in Probability Theory (ACMPT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10684))

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Abstract

In this paper we consider dynamics of complex systems using random neural networks with an infinite number of cells. The Cauchy problem for singular perturbed infinite order systems of stochastic differential equations which describes the random neural network with infinite number of cells is studied.

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References

  1. Calvert, B.D.: Neural networks with an infinite number of cells. J. Differ. Equ. 186(1), 31–51 (2002)

    Article  MathSciNet  Google Scholar 

  2. Calvert, B.D., Zemanian, A.H.: Operating points in infinite nonlinear networks approximated by finite networks. Trans. Am. Math. Soc. 352(2), 753–780 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Daletsky, Y.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. Science Publisher, Moscow (1970)

    Google Scholar 

  4. Gaidamaka, Y., Sopin, E., Talanova, M.: Approach to the analysis of probability measures of cloud computing systems with dynamic scaling. In: Vishnevsky, V., Kozyrev, D. (eds.) DCCN 2015. CCIS, vol. 601, pp. 121–131. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30843-2_13

    Chapter  Google Scholar 

  5. Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)

    Article  MATH  Google Scholar 

  6. Huang, H., Du, O., Kang, X.: Global exponential stability of neutral high-order stochastic Hopfield neural networks with Markovian jump parameters and mixed time delays. ISA Trans. 52(6), 759–767 (2013)

    Article  Google Scholar 

  7. Korobeinik, J.: Differential equations of infinite order and infinite systems of differential equations. Izv. Akad. Nauk SSSR Ser. Mat. 34, 881–922 (1970)

    MathSciNet  Google Scholar 

  8. Korolkova, A.V., Eferina, E.G., Laneev, E.B., Gudkova, I.A., Sevastianov, L.A., Kulyabov, D.S.: Stochastization of one-step processes in the occupations number representation. In: Proceedings - 30th European Conference on Modelling and Simulation, ECMS 2016, pp. 698–704 (2016)

    Google Scholar 

  9. Krasnoselsky, M.A., Zabreyko, P.P.: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984)

    Book  Google Scholar 

  10. Lomov, S.A.: The construction of asymptotic solutions of certain problems with parameters. Izv. Akad. Nauk SSSR Ser. Mat. 32, 884–913 (1968)

    MathSciNet  Google Scholar 

  11. Cho, M.W.: Competitive learning behavior in a stochastic neural network. J. Korean Phys. Soc. 67(9), 1679–1685 (2015)

    Article  Google Scholar 

  12. Persidsky, K.P.: Izv. AN KazSSR, Ser. Mat. Mach. 2, 3–34 (1948)

    Google Scholar 

  13. Samoilenko, A.M., Teplinskii, Y.V.: Countable Systems of Differential Equations. Brill, Leiden (2003)

    Book  MATH  Google Scholar 

  14. Samouylov, K., Naumov, V., Sopin, E., Gudkova, I., Shorgin, S.: Sojourn time analysis for processor sharing loss system with unreliable server. In: Wittevrongel, S., Phung-Duc, T. (eds.) ASMTA 2016. LNCS, vol. 9845, pp. 284–297. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43904-4_20

    Chapter  Google Scholar 

  15. Turchetti, C.: Stochastic Models of Neural Networks. Frontiers in Artificial Intelligence and Applications Knowledge-Based Intelligent Engineering Systems, vol. 102. IOS Press, Amsterdam (2004)

    MATH  Google Scholar 

  16. Turchetti, C., Crippa, P., Pirani, M., Biagetti, G.: Representation of nonlinear random transformations by non-Gaussian stochastic neural networks. IEEE Trans. Neural Netw. 19, 1033–1060 (2008). A Publication of the IEEE Neural Networks Council

    Article  Google Scholar 

  17. Tihonov, A.N.: Uber unendliche Systeme von Differentialgleichungen. Rec. Math. 41(4), 551–555 (1934)

    Google Scholar 

  18. Tihonov, A.N.: Systems of differential equations containing small parameters in the derivatives. Matematicheskii Sbornik (N.S.) 31(73), 575–586 (1952)

    MathSciNet  Google Scholar 

  19. Vasilyev, S.A., Kanzitdinov, S.K.: Model of neural networks with an infinite number of cells and small parameter. Int. Sci. J. Modern Inf. Technol. IT-Educ. 12(2), 15–20 (2016)

    Google Scholar 

  20. Vasil’eva, A.B.: Asymptotic behaviour of solutions of certain problems for ordinary non-linear differential equations with a small parameter multiplying the highest derivatives. Uspehi Mat. Nauk. 18(111(3)), 15–86 (1963)

    MathSciNet  Google Scholar 

  21. Zhautykov, O.A.: On a countable system of differential equations with variable parameters. Matematicheskii Sbornik (N.S.) 49(91), 317–330 (1959)

    MathSciNet  Google Scholar 

  22. Zhautykov, O.A.: Extension of the Hamilton-Jacobi theorems to an infinite canonical system of equations. Matematicheskii Sbornik (N.S.) 53(95), 313–328 (1961)

    MathSciNet  MATH  Google Scholar 

  23. Liang, X., Wang, L., Wang, Y., Wang, R.: Dynamical behavior of delayed reaction-diffusion Hopfield neural networks driven by infinite dimensional Wiener processes. IEEE Trans. Neural Netw. 27(9), 1816–1826 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgment

The publication was prepared with the support of the “RUDN University Program 5-100” and partially funded by RFBF grants No. 15-07-08795, No. 16-07-00556.

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Authors and Affiliations

  1. Department of Applied Probability and Informatics, RUDN University, Miklukho-Maklaya st. 6, Moscow, 117198, Russia

    Soltan K. Kanzitdinov & Sergey A. Vasilyev

Authors
  1. Soltan K. Kanzitdinov
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  2. Sergey A. Vasilyev
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Corresponding author

Correspondence to Sergey A. Vasilyev .

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Editors and Affiliations

  1. Department of Applied Probability and Informatics, RUDN University, Moscow, Russia

    Vladimir V. Rykov

  2. City University of Hong Kong, Hong Kong, Hong Kong

    Nozer D. Singpurwalla

  3. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Andrey M. Zubkov

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Kanzitdinov, S.K., Vasilyev, S.A. (2017). Infinite Order Systems of Differential Equations and Large Scale Random Neural Networks. In: Rykov, V., Singpurwalla, N., Zubkov, A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science(), vol 10684. Springer, Cham. https://doi.org/10.1007/978-3-319-71504-9_17

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  • DOI: https://doi.org/10.1007/978-3-319-71504-9_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71503-2

  • Online ISBN: 978-3-319-71504-9

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