Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

PARAMETRIZED GUDERMANNIAN FUNCTION RELIED BANACH SPACE VALUED NEURAL NETWORK MULTIVARIATE APPROXIMATIONS

  • GEORGE A. ANASTASSIOU (Department of Mathematical Sciences, University of Memphis)
  • Received : 2022.11.07
  • Accepted : 2023.02.15
  • Published : 2023.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

Here we give multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝN, N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a parametrized Gudermannian sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Keywords

References

  1. G.A. Anastassiou, Moments in Probability and Approximation Theory, Pitman Research Notes in Math., Vol. 287, Longman Sci. & Tech., Harlow, U.K., 1993.
  2. G.A. Anastassiou, Rate of convergence of some neural network operators to the unitunivariate case, J. Math. Anal. Appli. 212 (1997), 237-262. https://doi.org/10.1006/jmaa.1997.5494
  3. G.A. Anastassiou, Quantitative Approximations, Chapman&Hall/CRC, Boca Raton, New York, 2001.
  4. G.A. Anastassiou, Inteligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library, Vol. 19, Springer, Heidelberg, 2011.
  5. G.A. Anastassiou, Univariate hyperbolic tangent neural network approximation, Mathematics and Computer Modelling 53 (2011), 1111-1132. https://doi.org/10.1016/j.mcm.2010.11.072
  6. G.A. Anastassiou, Multivariate hyperbolic tangent neural network approximation, Computers and Mathematics 61 (2011), 809-821.
  7. G.A. Anastassiou, Multivariate sigmoidal neural network approximation, Neural Networks 24 (2011), 378-386. https://doi.org/10.1016/j.neunet.2011.01.003
  8. G.A. Anastassiou, Univariate sigmoidal neural network approximation, J. of Computational Analysis and Applications 14 (2012), 659-690.
  9. G.A. Anastassiou, Approximation by neural networks iterates, Advances in Applied Mathematics and Approximation Theory pp. 1-20, Springer Proceedings in Math. & Stat., Springer, New York, 2013.
  10. G.A. Anastassiou, Intelligent Systems II: Complete Approximation by Neural Network Operators, Springer, Heidelberg, New York, 2016.
  11. G.A. Anastassiou, Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg, New York, 2018.
  12. G.A. Anastassiou, General Multivariate arctangent function activated neural network approximations, J. Numer. Anal. Approx Theory 51 (2022), 37-66. https://doi.org/10.33993/jnaat511-1262
  13. G.A. Anastassiou, General sigmoid based Banach space valued neural network approximation, J. of Computational Analysis and Applications accepted, 2022.
  14. H. Cartan, Differential Calculus, Hermann, Paris, 1971.
  15. Z. Chen and F. Cao, The approximation operators with sigmoidal functions, Computers and Mathematics with Applications 58 (2009), 758-765. https://doi.org/10.1016/j.camwa.2009.05.001
  16. D. Costarelli, R. Spigler, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks 44 (2013), 101-106. https://doi.org/10.1016/j.neunet.2013.03.015
  17. D. Costarelli, R. Spigler, Multivariate neural network operators with sigmoidal activation functions, Neural Networks 48 (2013), 72-77. https://doi.org/10.1016/j.neunet.2013.07.009
  18. S. Haykin, Neural Networks: A Comprehensive Foundation,Prentice Hall, (2 ed.), New York, 1998.
  19. W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics 7 (1943), 115-133. https://doi.org/10.1007/BF02478259
  20. T.M. Mitchell, Machine Learning, WCB-McGraw-Hill, New York, 1997.
  21. L.B. Rall, Computational Solution of Nonlinear Operator Equations, John Wiley & Sons, New York, 1969.
  22. E.W. Weisstein, Gudermannian, MathWorld.