Multivariate error function based neural network approximations

George A. Anastassiou

doi:10.33993/jnaat431-992

Authors

George A. Anastassiou University of Memphis, USA

DOI:

https://doi.org/10.33993/jnaat431-992

Keywords:

error function, multivariate neural network approximation, quasi-interpolation operator, Baskakov type operator, quadrature type operator, multivariate modulus of continuity, complex approximation, iterated approximation

Abstract views: 287

Abstract

Here we present multivariate quantitative approximations of real and complex valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), by the multivariate quasi-interpolation, Baskakov type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last three 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 partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the Gaussian error special function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

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References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover Publications, 1972.

G. A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, J. Math. Anal. Appli., 212 (1997), pp. 237-262, https://doi.org/10.1006/jmaa.1997.5494 DOI: https://doi.org/10.1006/jmaa.1997.5494

G. A. Anastassiou, Quantitative Approximations, Chapman&Hall/CRC, Boca Raton, New York, 2001.

G. A. Anastassiou, Inteligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library, vol. 19, Springer, Heidelberg, 2011. DOI: https://doi.org/10.1007/978-3-642-21431-8

G. A. Anastassiou, Univariate hyperbolic tangent neural network approximation, Mathematics and Computer Modelling, 53 (2011), pp. 1111-1132, https://doi.org/10.1016/j.mcm.2010.11.072 DOI: https://doi.org/10.1016/j.mcm.2010.11.072

G. A. Anastassiou, Multivariate hyperbolic tangent neural network approximation, Computers and Mathematics 61 (2011), pp. 809-821, https://doi.org/10.1016/j.camwa.2010.12.029 DOI: https://doi.org/10.1016/j.camwa.2010.12.029

G. A. Anastassiou, Multivariate sigmoidal neural network approximation, Neural Networks 24 (2011), pp. 378-386, http://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art4 DOI: https://doi.org/10.1016/j.neunet.2011.01.003

G. A. Anastassiou, Univariate sigmoidal neural network approximation, J. of Computational Analysis and Applications, vol. 14 (2012), no. 4, pp. 659-690.

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, Eds. G. Anastassiou, O. Duman. DOI: https://doi.org/10.1007/978-1-4614-6393-1_1

G. A. Anastassiou, Univariate error function based neural network approximation, submitted, 2014. DOI: https://doi.org/10.1007/978-3-319-20505-2_17

L. C. Andrews, Special Functions of Mathematics for Engineers, Second edition, Mc Graw-Hill, New York, 1992.

Z. Chen and F. Cao, The approximation operators with sigmoidal functions, Computers and Mathematics with Applications, 58 (2009), pp. 758-765, https://doi.org/10.1016/j.camwa.2009.05.001 DOI: https://doi.org/10.1016/j.camwa.2009.05.001

D. Costarelli and R. Spigler, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks 44 (2013), pp. 101-106, https://doi.org/10.1016/j.neunet.2013.03.015 DOI: https://doi.org/10.1016/j.neunet.2013.03.015

D. Costarelli and R. Spigler, Multivariate neural network operators with sigmoidal activation functions, Neural Networks 48 (2013), pp. 72-77, https://doi.org/10.1016/j.neunet.2013.07.009 DOI: https://doi.org/10.1016/j.neunet.2013.07.009

S. Haykin, Neural Networks: A Comprehensive Foundation (2 ed.), Prentice Hall, New York, 1998.

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics, 7 (1943), pp. 115-133, http://ictp.acad.ro/jnaat/journal/article/view/1993-vol22-no1-art8 DOI: https://doi.org/10.1007/BF02478259

T. M. Mitchell, Machine Learning, WCB-McGraw-Hill, New York, 1997.