## Intelligent Systems: Approximation by Artificial Neural NetworksThis brief monograph is the first one to deal exclusively with the quantitative approximation by artificial neural networks to the identity-unit operator. Here we study with rates the approximation properties of the "right" sigmoidal and hyperbolic tangent artificial neural network positive linear operators. In particular we study the degree of approximation of these operators to the unit operator in the univariate and multivariate cases over bounded or unbounded domains. This is given via inequalities and with the use of modulus of continuity of the involved function or its higher order derivative. We examine the real and complex cases. For the convenience of the reader, the chapters of this book are written in a self-contained style. This treatise relies on author's last two years of related research work. Advanced courses and seminars can be taught out of this brief book. All necessary background and motivations are given per chapter. A related list of references is given also per chapter. The exposed results are expected to find applications in many areas of computer science and applied mathematics, such as neural networks, intelligent systems, complexity theory, learning theory, vision and approximation theory, etc. As such this monograph is suitable for researchers, graduate students, and seminars of the above subjects, also for all science libraries. |

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### Contents

1 | |

Univariate Hyperbolic Tangent Neural Network Quantitative Approximation | 33 |

Multivariate Sigmoidal NeuralNetwork Quantitative Approximation | 67 |

Multivariate Hyperbolic Tangent Neural Network Quantitative Approximation | 89 |

### Other editions - View all

Intelligent Systems: Approximation by Artificial Neural Networks George A. Anastassiou No preview available - 2011 |

Intelligent Systems: Approximation by Artificial Neural Networks George A. Anastassiou No preview available - 2013 |

### Common terms and phrases

a e a,b activation function Anastassiou Artificial Neural Networks assume further bounded functions chapter Chen Clearly f continuity w1 continuous and bounded continuous functions cosh defined engaged function F f,x Feed-forward Feed-forward neural networks FNNs further f give Theorem Hence nb hidden layer high order derivative hyperbolic tangent neural IN-T Intelligent Systems ISBN Jackson type inequalities k=|nal nb Let f e CB ma-k Mhaskar Micchelli modulus of continuity Multivariate Neural Network Network Quantitative Approximations neural network approximations neural network operators Neural Network Quantitative Order of approximation p(na present Theorem Proof proving the claim rate of convergence sigmoidal function sigmoidal neural network sins smooth functions Subcase Supremum norm tangent neural network tanh tanha tanhx Universal approximation w1 f wi f WTTW XL f XL q XL QP