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Multivariate numerical approximation using constructive \( L^{2} (\mathbb{R}) \) RBF neural network

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Abstract

For the multivariate continuous function, using constructive feedforward \( L^{2} (\mathbb{R}) \) radial basis function (RBF) neural network, we prove that a \( L^{2} (\mathbb{R}) \) RBF neural network with n + 1 hidden neurons can interpolate n + 1 multivariate samples with zero error. Then, we prove that the \( L^{2} (\mathbb{R}) \) RBF neural network can uniformly approximate any continuous multivariate function with arbitrary precision. The correctness and effectiveness are verified through eight numeric experiments.

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Acknowledgments

This work was supported by the Natural Science Foundation of China under Grants 10871208, 60970097 and 70871122, and was supported by the ZNDXQYYJJH under grant No. 2010QZZD015.

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Authors and Affiliations

  1. School of Mathematical Sciences and Computing Technology, Central South University, Changsha, 410083, China

    Hou Muzhou

  2. School of Information Science and Engineering, Central South University, Changsha, 410083, China

    Hou Muzhou & Han Xuli

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  1. Hou Muzhou
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  2. Han Xuli
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Correspondence to Hou Muzhou.

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Muzhou, H., Xuli, H. Multivariate numerical approximation using constructive \( L^{2} (\mathbb{R}) \) RBF neural network. Neural Comput & Applic 21, 25–34 (2012). https://doi.org/10.1007/s00521-011-0604-8

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  • DOI: https://doi.org/10.1007/s00521-011-0604-8

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