Fractional neural network approximation
Here, we study the univariate fractional quantitative approximation of real valued functions on a compact interval by quasi-interpolation sigmoidal and hyperbolic tangent neural network operators. These approximations are derived by establishing Jackson type inequalities involving the moduli of cont...
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| Published in | Computers & mathematics with applications (1987) Vol. 64; no. 6; pp. 1655 - 1676 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.09.2012
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| Abstract | Here, we study the univariate fractional quantitative approximation of real valued functions on a compact interval by quasi-interpolation sigmoidal and hyperbolic tangent neural network operators. These approximations are derived by establishing Jackson type inequalities involving the moduli of continuity of the right and left Caputo fractional derivatives of the engaged function. The approximations are pointwise and with respect to the uniform norm. The related feed-forward neural networks are with one hidden layer. Our fractional approximation results into higher order converges better than the ordinary ones. |
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| AbstractList | Here, we study the univariate fractional quantitative approximation of real valued functions on a compact interval by quasi-interpolation sigmoidal and hyperbolic tangent neural network operators. These approximations are derived by establishing Jackson type inequalities involving the moduli of continuity of the right and left Caputo fractional derivatives of the engaged function. The approximations are pointwise and with respect to the uniform norm. The related feed-forward neural networks are with one hidden layer. Our fractional approximation results into higher order converges better than the ordinary ones. |
| Author | Anastassiou, George A. |
| Author_xml | – sequence: 1 givenname: George A. surname: Anastassiou fullname: Anastassiou, George A. email: ganastss@memphis.edu, ganastss@gmail.com organization: Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA |
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| Keywords | Sigmoidal and hyperbolic tangent functions Neural network fractional approximation Modulus of continuity Quasi-interpolation operator Fractional derivative |
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| References | Anastassiou (br000005) 1997; 212 Anastassiou (br000010) 2001 Anastassiou (br000025) 2011; 53 Anastassiou (br000095) 2011; 54 Anastassiou (br000020) 2011; vol. 19 Haykin (br000045) 1998 Samko, Kilbas, Marichev (br000070) 1993 Anastassiou (br000090) 2009 G.A. Anastassiou, Univariate sigmoidal neural network approximation, Journal of Computational Analysis and Applications (2011) (in press). Chen, Cao (br000015) 2009; 58 Diethelm (br000060) 2010 Anastassiou (br000035) 2011; 24 Frederico, Torres (br000065) 2008; 3 Anastassiou (br000075) 2009; 42 Anastassiou (br000080) 2009; 42 El-Sayed, Gaber (br000085) 2006; 3 Mitchell (br000055) 1997 Anastassiou (br000030) 2011; 61 McCulloch, Pitts (br000050) 1943; 7 Anastassiou (10.1016/j.camwa.2012.01.019_br000030) 2011; 61 Anastassiou (10.1016/j.camwa.2012.01.019_br000095) 2011; 54 McCulloch (10.1016/j.camwa.2012.01.019_br000050) 1943; 7 Mitchell (10.1016/j.camwa.2012.01.019_br000055) 1997 El-Sayed (10.1016/j.camwa.2012.01.019_br000085) 2006; 3 Frederico (10.1016/j.camwa.2012.01.019_br000065) 2008; 3 Anastassiou (10.1016/j.camwa.2012.01.019_br000020) 2011; vol. 19 Haykin (10.1016/j.camwa.2012.01.019_br000045) 1998 Anastassiou (10.1016/j.camwa.2012.01.019_br000035) 2011; 24 Chen (10.1016/j.camwa.2012.01.019_br000015) 2009; 58 10.1016/j.camwa.2012.01.019_br000040 Anastassiou (10.1016/j.camwa.2012.01.019_br000075) 2009; 42 Anastassiou (10.1016/j.camwa.2012.01.019_br000080) 2009; 42 Anastassiou (10.1016/j.camwa.2012.01.019_br000090) 2009 Diethelm (10.1016/j.camwa.2012.01.019_br000060) 2010 Anastassiou (10.1016/j.camwa.2012.01.019_br000005) 1997; 212 Anastassiou (10.1016/j.camwa.2012.01.019_br000010) 2001 Anastassiou (10.1016/j.camwa.2012.01.019_br000025) 2011; 53 Samko (10.1016/j.camwa.2012.01.019_br000070) 1993 |
| References_xml | – volume: 24 start-page: 378 year: 2011 end-page: 386 ident: br000035 article-title: Multivariate sigmoidal neural network approximation publication-title: Neural Networks contributor: fullname: Anastassiou – volume: 3 start-page: 81 year: 2006 end-page: 95 ident: br000085 article-title: On the finite Caputo and finite Riesz derivatives publication-title: Electronic Journal of Theoretical Physics contributor: fullname: Gaber – volume: 3 start-page: 479 year: 2008 end-page: 493 ident: br000065 article-title: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem publication-title: International Mathematical Forum contributor: fullname: Torres – volume: 54 start-page: 3098 year: 2011 end-page: 3115 ident: br000095 article-title: Fractional representation formulae and right fractional inequalities publication-title: Mathematical and Computer Modelling contributor: fullname: Anastassiou – year: 2009 ident: br000090 article-title: Fractional Differentiation Inequalities contributor: fullname: Anastassiou – year: 2001 ident: br000010 article-title: Quantitative Approximations contributor: fullname: Anastassiou – volume: 212 start-page: 237 year: 1997 end-page: 262 ident: br000005 article-title: Rate of convergence of some neural network operators to the unit-univariate case publication-title: Journal of Mathematical Analysis and Applications contributor: fullname: Anastassiou – volume: 7 start-page: 115 year: 1943 end-page: 133 ident: br000050 article-title: A logical calculus of the ideas immanent in nervous activity publication-title: Bulletin of Mathematical Biophysics contributor: fullname: Pitts – volume: vol. 19 year: 2011 ident: br000020 publication-title: Inteligent Systems: Approximation by Artificial Neural Networks contributor: fullname: Anastassiou – volume: 58 start-page: 758 year: 2009 end-page: 765 ident: br000015 article-title: The approximation operators with sigmoidal functions publication-title: Computers and Mathematics with Applications contributor: fullname: Cao – volume: 61 start-page: 809 year: 2011 end-page: 821 ident: br000030 article-title: Multivariate hyperbolic tangent neural network approximation publication-title: Computer and Mathematics contributor: fullname: Anastassiou – year: 1997 ident: br000055 article-title: Machine Learning contributor: fullname: Mitchell – year: 1993 ident: br000070 article-title: Fractional Integrals and Derivatives, Theory and Applications contributor: fullname: Marichev – volume: 42 start-page: 2080 year: 2009 end-page: 2094 ident: br000075 article-title: Fractional Korovkin theory publication-title: Chaos, Solitons & Fractals contributor: fullname: Anastassiou – year: 1998 ident: br000045 article-title: Neural Networks: A Comprehensive Foundation contributor: fullname: Haykin – volume: 53 start-page: 1111 year: 2011 end-page: 1132 ident: br000025 article-title: Univariate hyperbolic tangent neural network approximation publication-title: Mathematical and Computer Modelling contributor: fullname: Anastassiou – year: 2010 ident: br000060 publication-title: The Analysis of Fractional Differential Equations contributor: fullname: Diethelm – volume: 42 start-page: 365 year: 2009 end-page: 376 ident: br000080 article-title: On right fractional calculus publication-title: Chaos, Solitons & Fractals contributor: fullname: Anastassiou – volume: 53 start-page: 1111 year: 2011 ident: 10.1016/j.camwa.2012.01.019_br000025 article-title: Univariate hyperbolic tangent neural network approximation publication-title: Mathematical and Computer Modelling doi: 10.1016/j.mcm.2010.11.072 contributor: fullname: Anastassiou – volume: vol. 19 year: 2011 ident: 10.1016/j.camwa.2012.01.019_br000020 contributor: fullname: Anastassiou – ident: 10.1016/j.camwa.2012.01.019_br000040 doi: 10.1007/978-3-642-21431-8_1 – volume: 58 start-page: 758 year: 2009 ident: 10.1016/j.camwa.2012.01.019_br000015 article-title: The approximation operators with sigmoidal functions publication-title: Computers and Mathematics with Applications doi: 10.1016/j.camwa.2009.05.001 contributor: fullname: Chen – volume: 7 start-page: 115 year: 1943 ident: 10.1016/j.camwa.2012.01.019_br000050 article-title: A logical calculus of the ideas immanent in nervous activity publication-title: Bulletin of Mathematical Biophysics doi: 10.1007/BF02478259 contributor: fullname: McCulloch – year: 1997 ident: 10.1016/j.camwa.2012.01.019_br000055 contributor: fullname: Mitchell – year: 2010 ident: 10.1016/j.camwa.2012.01.019_br000060 contributor: fullname: Diethelm – volume: 3 start-page: 479 issue: 10 year: 2008 ident: 10.1016/j.camwa.2012.01.019_br000065 article-title: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem publication-title: International Mathematical Forum contributor: fullname: Frederico – volume: 24 start-page: 378 year: 2011 ident: 10.1016/j.camwa.2012.01.019_br000035 article-title: Multivariate sigmoidal neural network approximation publication-title: Neural Networks doi: 10.1016/j.neunet.2011.01.003 contributor: fullname: Anastassiou – volume: 61 start-page: 809 year: 2011 ident: 10.1016/j.camwa.2012.01.019_br000030 article-title: Multivariate hyperbolic tangent neural network approximation publication-title: Computer and Mathematics contributor: fullname: Anastassiou – volume: 212 start-page: 237 year: 1997 ident: 10.1016/j.camwa.2012.01.019_br000005 article-title: Rate of convergence of some neural network operators to the unit-univariate case publication-title: Journal of Mathematical Analysis and Applications doi: 10.1006/jmaa.1997.5494 contributor: fullname: Anastassiou – volume: 42 start-page: 2080 issue: 4 year: 2009 ident: 10.1016/j.camwa.2012.01.019_br000075 article-title: Fractional Korovkin theory publication-title: Chaos, Solitons & Fractals doi: 10.1016/j.chaos.2009.03.183 contributor: fullname: Anastassiou – year: 1998 ident: 10.1016/j.camwa.2012.01.019_br000045 contributor: fullname: Haykin – year: 2001 ident: 10.1016/j.camwa.2012.01.019_br000010 contributor: fullname: Anastassiou – year: 1993 ident: 10.1016/j.camwa.2012.01.019_br000070 contributor: fullname: Samko – volume: 42 start-page: 365 year: 2009 ident: 10.1016/j.camwa.2012.01.019_br000080 article-title: On right fractional calculus publication-title: Chaos, Solitons & Fractals doi: 10.1016/j.chaos.2008.12.013 contributor: fullname: Anastassiou – year: 2009 ident: 10.1016/j.camwa.2012.01.019_br000090 contributor: fullname: Anastassiou – volume: 54 start-page: 3098 issue: 11–12 year: 2011 ident: 10.1016/j.camwa.2012.01.019_br000095 article-title: Fractional representation formulae and right fractional inequalities publication-title: Mathematical and Computer Modelling doi: 10.1016/j.mcm.2011.07.040 contributor: fullname: Anastassiou – volume: 3 start-page: 81 issue: 12 year: 2006 ident: 10.1016/j.camwa.2012.01.019_br000085 article-title: On the finite Caputo and finite Riesz derivatives publication-title: Electronic Journal of Theoretical Physics contributor: fullname: El-Sayed |
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| SubjectTerms | Approximation Fractional derivative Modulus of continuity Neural network fractional approximation Quasi-interpolation operator Sigmoidal and hyperbolic tangent functions |
| Title | Fractional neural network approximation |
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