×

zbMATH — the first resource for mathematics

Approximation by neural networks with sigmoidal functions. (English) Zbl 1311.41015
Summary: In this paper, we introduce a type of approximation operators of neural networks with sigmodal functions on compact intervals, and obtain the pointwise and uniform estimates of the approximation. To improve the approximation rate, we further introduce a type of combinations of neural networks. Moreover, we show that the derivatives of functions can also be simultaneously approximated by the derivatives of the combinations. We also apply our method to construct approximation operators of neural networks with sigmodal functions on infinite intervals.

MSC:
41A25 Rate of convergence, degree of approximation
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
92B20 Neural networks for/in biological studies, artificial life and related topics
PDF BibTeX Cite
Full Text: DOI
References:
[1] Anastassiou, G. A.: Rate of convergence of some neural network operators to the unit-univariate case. J. Math. Anal. Appl., 212, 237–262 (1997) · Zbl 0899.68088
[2] Barron, A. R.: Universal approximation bounds for superpositions of a sigmodal function. IEEE Trans. Inform. Theory, 39, 930–945 (1993) · Zbl 0818.68126
[3] Cao, F.-L., Xie, T.-F., Xu, Z.-B.: The estimate for approximation error of neural networks: A constructive approach. Neurocomputing, 71, 626–636 (2008) · Zbl 05718504
[4] Chen, T.-P., Chen, H.: Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to a dynamic system. IEEE Trans. Neural Netw., 6, 911–917 (1995)
[5] Chen, Z.-X., Cao, F.-L., The approximation operators with sigmoidal functions. Comput. Math. Appl., 58, 758–765 (2009) · Zbl 1189.41014
[6] Chui, C.-K., Li, X.: Approximation by ridge functions and neural networks with one hidden layer. J. Approx. Theory, 70, 131–141 (1992) · Zbl 0768.41018
[7] Cybenko, G.: Approximation by superpositions of sigmoidal function. Math. Control Signals Sys., 2, 303–314 (1989) · Zbl 0679.94019
[8] Ferrari, S., Stengel, R. F.: Smooth function approximation using neural networks. IEEE Trans. Neural Netw., 16, 24–38 (2005)
[9] Funahashi, K. I.: On the approximate realization of continuous mapping by neural networks. Neural Netw., 2, 183–192 (1989)
[10] Hahm, N., Hong, B. I.: An approximation by neural networks with a fixed weight. Comput. Math. Appl., 47, 1897–1903 (2004) · Zbl 1065.41047
[11] Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximation. Neural Netw., 2, 359–366 (1989) · Zbl 1383.92015
[12] Hornik, K., Stinchcombe, M., White, H.: Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw., 3, 551–560 (1990)
[13] Leshno, M., Lin, V. Y., Pinks, A., et al.: Multilayer feedward networks with a nonpolynomial activation function can approximate any function. Neural Netw., 6, 861–867 (1993)
[14] Lorentz, G. G.: Approximation of Functions, Holt, Rinehart and Winston, New York, 1966 · Zbl 0153.38901
[15] Maiorov, V., Meir, R. S.: Approximation bounds for smooth functions in C(\(\mathbb{R}\)d) by neural and moxture networks. IEEE Trans. Neural Netw., 9, 969–978 (1998)
[16] Makovoz, Y.: Uinform approximation by neural networks. J. Approx. Theory, 95, 215–228 (1998) · Zbl 0932.41016
[17] Mhaskar, H. N., Micchelli, C. A.: Approximation by superposition of a sigmodal function. Adv. Appl. Math., 13, 350–373 (1992) · Zbl 0763.41015
[18] Mhaskar, H. N., Micchelli, C. A.: Degree of approximation by neural networks with a single hidden layer. Adv. Appl. Math., 16, 151–183 (1995) · Zbl 0885.42012
[19] Ramazanov, A. R. K.: On approximation by polynomials and rational functions in Orlicz spaces. Anal. Math., 10, 117–132 (1984) · Zbl 0566.41015
[20] Xu, Z.-B., Cao, F.-L.: The essential order of approximation for neural networks. Sci. China, Ser. F, 47, 97–112 (2004) · Zbl 1186.82063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.