Skip to main content
SpringerLink
Log in

q-Deformed Hyperbolic Tangent Relied Banach Space Valued Multivariate Multi Layer Neural Network Approximation

  • Conference paper
  • First Online:
Advances in Mathematical Modelling, Applied Analysis and Computation (ICMMAAC 2023)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 953))

  • 28 Accesses

Abstract

Here we study the multivariate quantitative approximation of Banach space valued continuous multivariate functions on a box or \(\mathbb {R}^{N}\), \(N\in \mathbb {N},\) by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We research also the case of approximation by iterated multilayer neural network operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives or partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed hyperbolic tangent sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anastassiou, G.A.: Moments in Probability and Approximation Theory. Pitman Research Notes in Mathematics, vol. 287. Longman Scientific & Technical, Harlow (1993)

    Google Scholar 

  2. Anastassiou, G.A.: Rate of convergence of some neural network operators to the unit-univariate case. J. Math. Anal. Appl. 212, 237–262 (1997)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Book  Google Scholar 

  5. Anastassiou, G.A.: Univariate hyperbolic tangent neural network approximation. Math. Comput. Model. 53, 1111–1132 (2011)

    Article  MathSciNet  Google Scholar 

  6. Anastassiou, G.A.: Multivariate hyperbolic tangent neural network approximation. Comput. Math. 61, 809–821 (2011)

    MathSciNet  Google Scholar 

  7. Anastassiou, G.A.: Multivariate sigmoidal neural network approximation. Neural Netw. 24, 378–386 (2011)

    Article  Google Scholar 

  8. Anastassiou, G.A.: Univariate sigmoidal neural network approximation. J. Comput. Anal. Appl. 14(4), 659–690 (2012)

    MathSciNet  Google Scholar 

  9. Anastassiou, G.A.: Approximation by neural networks iterates. In: Anastassiou, G., Duman, O. (eds.) Advances in Applied Mathematics and Approximation Theory. Springer Proceedings in Mathematics & Statistics, vol. 41, pp. 1–20. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-6393-1_1

    Chapter  Google Scholar 

  10. Anastassiou, G.: Intelligent Systems II: Complete Approximation by Neural Network Operators. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-20505-2

    Book  Google Scholar 

  11. Anastassiou, G.: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3-319-66936-6

    Book  Google Scholar 

  12. Anastassiou, G.A.: Banach Space Valued Neural Network. Springer, Heidelberg (2023). https://doi.org/10.1007/978-3-031-16400-2

    Book  Google Scholar 

  13. Anastassiou, G.A.: \(q\)-Deformed hyperbolic tangent based Banach space valued ordinary and fractional neural network approximations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 83 (2023)

    Article  MathSciNet  Google Scholar 

  14. Cartan, H.: Differential Calculus. Hermann, Paris (1971)

    Google Scholar 

  15. Chen, Z., Cao, F.: The approximation operators with sigmoidal functions. Comput. Math. Appl. 58, 758–765 (2009)

    Article  MathSciNet  Google Scholar 

  16. Costarelli, D., Spigler, R.: Approximation results for neural network operators activated by sigmoidal functions. Neural Netw. 44, 101–106 (2013)

    Article  Google Scholar 

  17. Costarelli, D., Spigler, R.: Multivariate neural network operators with sigmoidal activation functions. Neural Netw. 48, 72–77 (2013)

    Article  Google Scholar 

  18. El-Shehawy, S.A., Abdel-Salam, E.A.-B.: The \(q\)-deformed hyperbolic Secant family. Int. J. Appl. Math. Stat. 29(5), 51–62 (2012)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  20. McCulloch, W., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 7, 115–133 (1943)

    Article  MathSciNet  Google Scholar 

  21. Mitchell, T.M.: Machine Learning. WCB-McGraw-Hill, New York (1997)

    Google Scholar 

  22. Rall, L.B.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    George A. Anastassiou

Authors
  1. George A. Anastassiou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George A. Anastassiou .

Editor information

Editors and Affiliations

  1. Department of Mathematics, JECRC University, Jaipur, Rajasthan, India

    Jagdev Singh

  2. Department of Mathematics, University of Memphis, Memphis, TN, USA

    George A. Anastassiou

  3. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Dumitru Baleanu

  4. Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan, India

    Devendra Kumar

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Anastassiou, G.A. (2024). q-Deformed Hyperbolic Tangent Relied Banach Space Valued Multivariate Multi Layer Neural Network Approximation. In: Singh, J., Anastassiou, G.A., Baleanu, D., Kumar, D. (eds) Advances in Mathematical Modelling, Applied Analysis and Computation . ICMMAAC 2023. Lecture Notes in Networks and Systems, vol 953. Springer, Cham. https://doi.org/10.1007/978-3-031-56304-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-56304-1_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-56303-4

  • Online ISBN: 978-3-031-56304-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation