Doi: http://dx.doi.org/10.15446/recolma.v49n1.54179

¹University of Memphis, Memphis, U.S.A. Email: ganastss@memphis.edu

Here we examine the quasi-interpolation error function based neural network operators of one hidden layer. Based on fractional calculus theory we derive a fractional Voronovskaya type asymptotic expansion for the error of approximation of these operators to the unit operator, as we are studying the univariate case. We treat also analogously the multivariate case.

Key words: Neural Network Fractional Approximation, Multivariate Neural Network Approximation, Voronovskaya AsymptoticExpansions, Fractional derivative, Error function.

Aquí se examinan funciones de error basadas en cuasi-interpolación de operadores de redes neuronales de una capa oculta. Basado en teoría de cálculo fraccional se deriva una expansión de asintótica de tipo Voronovskaya para el error de aproximación de estos operadores al operador unitario, así como el caso univariado. También se trata análogamente el caso multivariado.

Palabras clave: Aproximación fraccional de redes neuronales, expansión asintótica de Voronovskaya, derivada fraccional, función error.

Texto completo disponible en PDF

References

[1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, USA, 1972.         [ Links ]

[2] G. A. Anastassiou, `Rate of Convergence of Some Neural Network Operators to the Unit-Univariate Case´, J. Math. Anal. Appli. 212, (1997), 237-262.         [ Links ]

[3] G. A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, New York, USA, 2001.         [ Links ]

[4] G. A. Anastassiou, `Fractional Korovkin Theory´, Chaos, Solitons & Fractals 42, 4 (2009a), 2080-2094.         [ Links ]

[5] G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, New York, USA, 2009b.         [ Links ]

[6] G. A. Anastassiou, `On Right Fractional Calculus´, Chaos, Solitons & Fractals 42, 1 (2009b), 365-376.         [ Links ]

[7] G. A. Anastassiou, `Multivariate Sigmoidal Neural Network Approximation´, Neural Networks 24, (2011a), 378-386.         [ Links ]

[8] G. A. Anastassiou, `Multivariate Hyperbolic Tangent Neural Network Approximation´, Computers and Mathematics 61, (2011b), 809-821.         [ Links ]

[9] G. A. Anastassiou, `Univariate Hyperbolic Tangent Neural Network Approximation´, Mathematics and Computer Modelling 53, (2011b), 1111-1132.         [ Links ]

[10] G. A. Anastassiou, Advanced Inequalities, World Scientific Publ. Co., New Jersey, USA, 2011b.         [ Links ]

[11] G. A. Anastassiou, `Fractional Representation Formulae and Right Fractional Inequalities´, Mathematical and Computer Modelling 54, 11-12 (2011b), 3098-3115.         [ Links ]

[12] G. A. Anastassiou, Inteligent Systems: Approximation by Artificial Neural Networks, `Intelligent Systems Reference Library´, (2011b), Vol. 19, Springer, Heidelberg, Germany.         [ Links ]

[13] G. A. Anastassiou, `Fractional Neural Network Approximation´, Computers and Mathematics with Applications 64, (2012a), 1655-1676.         [ Links ]

[14] G. A. Anastassiou, `Univariate Sigmoidal Neural Network Approximation´, Journal of Computational Analysis and Applications 14, 4 (2012b), 659-690.         [ Links ]

[15] G. A. Anastassiou, Multivariate Error Function Based Neural Network Approximations, submitted, 2014a.         [ Links ]

[16] G. A. Anastassiou, Univariate Error Function Based Neural Network Approximation, submitted, 2014b.         [ Links ]

[17] L. C. Andrews, Special Functions of Mathematics for Engineers, Second edn, Mc Graw-Hill, New York, USA, 1992.         [ Links ]

[18] Z. Chen and F. Cao, `The Approximation Operators With Sigmoidal Functions´, Computers and Mathematics with Applications 58, (2009), 758-765.         [ Links ]

[19] D. Costarelli and R. Spigler, `Multivariate Neural Network Operators With Sigmoidal Activation Functions´, Neural Networks 48, (2013a), 72-77.         [ Links ]

[20] D. Costarelli and R. Spigler, `Approximation Results for Neural Network Operators Activated by Sigmoidal Functions´, Neural Networks 44, (2013b), 101-106.         [ Links ]

[21] K. Diethelm, The Analysis of Fractional Differential Equations, `Lecture Notes in Mathematics 2004´, (2010), Springer-Verlag, Berlin, Germany.         [ Links ]

[22] A. M. A. El-Sayed and M. Gaber, `On the Finite Caputo and Finite Riesz Derivatives´, Electronic Journal of Theoretical Physics 3, 12 (2006), 81-95.         [ Links ]

[23] G. S. Frederico and D. F. M. Torres, `Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem´, International Mathematical Forum 3, 10 (2008), 479-493.         [ Links ]

[24] S. Haykin, Neural Networks: A Comprehensive Foundation, Second edn, Prentice Hall, New York, USA, 1998.         [ Links ]

[25] W. McCulloch and W. Pitts, `A Logical Calculus of the Ideas Immanent in Nervous Activity´, Bulletin of Mathematical Biophysics 7, (1943), 115-133.         [ Links ]

[26] T. M. Mitchell, Machine Learning, WCB-McGraw-Hill, New York, USA, 1997.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

Doi: http://dx.doi.org/ @ARTICLE{RCMv49n1a09,
    AUTHOR  = {Anastassiou, George A.},
    TITLE   = {{Voronovskaya Type Asymptotic Expansions for Error Function Based Quasi-Interpolation Neural Network Operators}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2015},
    volume  = {49},
    number  = {1},
    pages   = {171--192}
}