Fuzzy Ordinary and Fractional General Sigmoid Function Activated Neural Network Approximation
Subject Areas : Transactions on Fuzzy Sets and Systems
1 - Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A.
Keywords: General sigmoid activation function, Neural network fuzzy fractional approximation, Fuzzy quasi-interpolation operator, Fuzzy modulus of continuity, Fuzzy derivative and fuzzy fractional derivative,
Abstract :
Here we research the univariate fuzzy ordinary and fractional quantitative approximation of fuzzy real valued functions on a compact interval by quasi-interpolation general sigmoid activation function relied on fuzzy neural network operators. These approximations are derived by establishing fuzzy Jackson type inequalities involving the fuzzy moduli of continuity of the function, or of the right and left Caputo fuzzy fractional derivatives of the involved function. The approximations are fuzzy pointwise and fuzzy uniform. The related feed-forward fuzzy neural networks are with one hidden layer. We study in particular the fuzzy integer derivative and just fuzzy continuous cases. Our fuzzy fractional approximation result using higher order fuzzy differentiation converges better than in the fuzzy just continuous case.
[1] G. A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, Journal of Mathematical Analysis and Application, 212 (1997), 237-262.
[2] G. A. Anastassiou, Quantitative Approximation, Chapmann and Hall/CRC, Boca Raton, New York, (2001).
[3] G. A. Anastassiou, Fuzzy approximation by fuzzy convolution type operators, Computers and Mathematics, 48 (2004), 1369-1386.
[4] G. A. Anastassiou, Higher order fuzzy korovkin theory via inequalities, Communications in Applied Analysis, 10(2) (2006), 359-392.
[5] G. A. Anastassiou, Fuzzy korovkin theorems and inequalities, Journal of Fuzzy Mathematics, 15(1) (2007), 169-205.
[6] G. A. Anastassiou, On right fractional calculus, Chaos, solitons and fractals, 42 (2009), 365-376.
[7] G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, New York, (2009).
[8] G. A. Anastassiou, Fractional korovkin theory, Chaos, Solitons & Fractals, 42(4) (2009), 2080-2094.
[9] G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory, Springer, Heildelberg, New York, (2010).
[10] G. A. Anastassiou, Intelligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library, Springer, Heidelberg, 19 (2011).
[11] G. A. Anastassiou, Fuzzy fractional calculus and ostrowski inequality, J. Fuzzy Math., 19(3) (2011), 577-590.
[12] G. A. Anastassiou, Fractional representation formulae and right fractional inequalities, Mathematical and Computer Modelling, 54(11-12) (2011), 3098-3115.
[13] G. A. Anastassiou, Univariate hyperbolic tangent neural network approximation, Mathematics and Computer Modelling, 53 (2011), 1111-1132.
[14] G. A. Anastassiou, Multivariate hyperbolic tangent neural network approximation, Computers and Mathematics, 61 (2011), 809-821.
[15] G. A. Anastassiou, Multivariate sigmoidal neural network approximation, Neural Networks, 24 (2011), 378-386.
[16] G. A. Anastassiou, Univariate sigmoidal neural network approximation, J. of Computational Analysis and Applications, 14(4) (2012), 659-690.
[17] G. A. Anastassiou, Fractional neural network approximation, Computers and Mathematics with Applications, 64(6) (2012), 1655-1676.
[18] G. A. Anastassiou, Fuzzy fractional neural network approximation by fuzzy quasi-interpolation operators, J. of Applied Nonlinear Dynamics, 2(3) (2013), 235-259.
[19] G. A. Anastassiou, Intelligent Systems II: Complete Approximation by Neural Network Operators, Springer, Heidelberg, New York, (2016).
[20] G. A. Anastassiou, Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg, New York, (2018).
[21] G. A. Anastassiou, Banach Space Valued Neural Network, Springer, Heidelberg, New York, (2023). [22] G. A. Anastassiou, General sigmoid based Banach space valued neural network approximation, J. Computational Analysis and Applications, 31(4) (2023), 520-534.
[23] Z. Chen and F. Cao, The approximation operators with sigmoidal functions, Computers and Mathematics with Applications, 58 (2009), 758-765.
[24] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics 2004, Springer-Verlag, Berlin, Heidelberg, (2010).
[25] 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.
[26] 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.
[27] S. Gal, Approximation Theory in Fuzzy Setting, Chapter 13 in Handbook of Analytic-Computational Methods in Applied Mathematics, 617-666, edited by G. Anastassiou, Chapman & Hall/CRC, Boca Raton, New York, (2000).
[28] R. Goetschel Jr., W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43.
[29] S. Haykin, Neural Networks: A Comprehensive Foundation (2 ed.), Prentice Hall, New York, (1998).
[30] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
[31] Y. K. Kim, B. M. Ghil, Integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems, 86 (1997), 213-222.
[32] W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics, 7 (1943), 115-133.
[33] T. M. Mitchell, Machine Learning, WCB-McGraw-Hill, New York, (1997).
[34] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, (1993), (English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk, (1987)).
[35] C. Wu, Z. Gong, On Henstock integrals of interval-valued functions and fuzzy valued functions, Fuzzy Sets and Systems, 115(3) (2000), 377-391.
[36] C. Wu, Z. Gong, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Sets and Systems, 120(3) (2001), 523-532.
[37] C. Wu, M. Ma, On embedding problem of fuzzy numer spaces: Part 1, Fuzzy Sets and Systems, 44 (1991), 33-38.