Skip to main content
SpringerLink
Log in

The construction and approximation of feedforward neural network with hyperbolic tangent function

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tangent activation function. Firstly, an equation of partitions of unity for the hyperbolic tangent function is given. Then, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectively. Also, the errors of the approximation are estimated by means of the modulus of continuity of function. Moreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  3. G A Anastassiou. Univariate hyperbolic tangent neural network approximation, Math Comput Modelling, 2011, 53: 1111–1132.

    MATH  MathSciNet  Google Scholar 

  4. G A Anastassiou. Multivariate hyperbolic tangent neural network approximation, Comput Math Appl, 2011, 61: 809–821.

    MATH  MathSciNet  Google Scholar 

  5. G A Anastassiou. Multivariate sigmoidal neural network approximation, Neural Networks, 2011, 24: 378–386.

    Article  MATH  Google Scholar 

  6. G A Anastassiou. Intelligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library 19, Berlin, Springer-Verlag, 2011.

    Book  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  8. A R Barron. Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans Inform Theory, 1993, 39: 930–945.

    MATH  MathSciNet  Google Scholar 

  9. F L Cao, T F Xie, Z B Xu. The estimate for approximation error of neural networks: A constructive approach, Neurocomputing, 2008, 71: 626–630.

    Google Scholar 

  10. D B Chen. Degree of approximation by superpositions of a sigmoidal function, Approx Theory Appl, 1993, 9: 17–28.

    MATH  Google Scholar 

  11. T P Chen, H Chen. Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks, IEEE Trans Neural Networks, 1995, 6: 904–910.

    Google Scholar 

  12. T P Chen, H Chen. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to a dynamic system, IEEE Trans Neural Networks, 1995, 6: 911–917.

    Google Scholar 

  13. T P Chen, H Chen, R W &Liu. Approximation capability in C(Rn) by multilayer feedforward networks and related problems, IEEE Trans Neural Networks, 1995, 6: 25–30.

    MATH  Google Scholar 

  14. Z X Chen, F L Cao. The approximation operators with sigmoidal functions, Comput Math Appl, 2009, 58: 758–765.

    MATH  MathSciNet  Google Scholar 

  15. C K Chui, X Li. Approximation by ridge functions and neural networks with one hidden layer, J Approx Theory, 1992, 70: 131–141.

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  17. D Costarelli, R Spigler. Constructive approximation by superposition of sigmoidal functions, Anal Theory Appl, 2013, 29: 169–196.

    MATH  MathSciNet  Google Scholar 

  18. D Costarelli, R Spigler. Multivarite neural network operators with sigmoidal activation functions, Neural Networks, 2013, 48: 72–77.

    MATH  Google Scholar 

  19. D Costarelli. Interpolation by neural network operators activated by ramp functions, JMath Anal Appl, 2014, 419: 574–582.

    MATH  MathSciNet  Google Scholar 

  20. D Costarelli, R Spigler. Convergence of a family of neural network operators of the Kantorovich type, J Approx Theory, 2014, 185: 80–90.

    MATH  MathSciNet  Google Scholar 

  21. G Cybenko. Approximation by superpositions of sigmoidal function, Math Control Signals Systems, 1989, 2: 303–314.

    MATH  MathSciNet  Google Scholar 

  22. K I Funahashi. On the approximaterealization of continuous mappings by neural networks, Neural Networks, 1989, 2: 183–192.

    Article  Google Scholar 

  23. B Gao, Y Xu. Univariate approximation by superpositions of a sigmoidal function, J Math Anal Appl, 1993, 178: 221–226.

    MATH  MathSciNet  Google Scholar 

  24. K Hornik, M Stinchcombe, H White. Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks, Neural Networks, 1990, 3: 551–560.

    Google Scholar 

  25. P C Kainen, V Kurkov´a. An integral upper bound for neural network approximation, Neural Comput, 2009, 21: 2970–2989.

    MathSciNet  Google Scholar 

  26. Y Makovoz. Uniform approximation by neural networks, J Approx Theory, 1998, 95: 215–228.

    MATH  MathSciNet  Google Scholar 

  27. D S Mitrinovic. Analytic Inequalities, Springer-Verlag, 1970.

    Book  MATH  Google Scholar 

  28. A Pinkus. Approximation theory of the MLP model in neural networks, Acta Numer, 1999, 8: 143–195.

    MathSciNet  Google Scholar 

  29. S Suzuki. Constructive function approximation by three-layer neural networks, Neural Networks, 1998, 11: 1049–1058.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shaoxing University, Shaoxing, 312000, China

    Zhi-xiang Chen

  2. Department of Mathematics, China Jiliang University, Hangzhou, 310018, China

    Fei-long Cao

Authors
  1. Zhi-xiang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fei-long Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei-long Cao.

Additional information

Supported by the National Natural Science Foundation of China (61179041, 61272023, and 11401388).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Zx., Cao, Fl. The construction and approximation of feedforward neural network with hyperbolic tangent function. Appl. Math. J. Chin. Univ. 30, 151–162 (2015). https://doi.org/10.1007/s11766-015-3000-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-015-3000-9

MR Subject Classification

Keywords

Search

Navigation