Banach Space Valued Neural Network: Ordinary and Fractional Approximation and Interpolation (Studies in Computational Intelligence, 1062)

Description

This book is about the generalization and modernization of approximation by neural network operators. Functions under approximation and the neural networks are Banach space valued. These are induced by a great variety of activation functions deriving from the arctangent, algebraic, Gudermannian, and generalized symmetric sigmoid functions. Ordinary, fractional, fuzzy, and stochastic approximations are exhibited at the univariate, fractional, and multivariate levels. Iterated-sequential approximations are also covered. The book’s results are expected to find applications in the many areas of applied mathematics, computer science and engineering, especially in artificial intelligence and machine learning. Other possible applications can be in applied sciences like statistics, economics, etc. Therefore, this book is suitable for researchers, graduate students, practitioners, and seminars of the above disciplines, also to be in all science and engineering libraries. Preface Contents 1 Algebraic Function Induced Banach Space Valued Ordinary and Fractional Neural Network Approximations 1.1 Introduction 1.2 Basics 1.3 Main Results References 2 Gudermannian Function Induced Banach Space Valued Ordinary and Fractional Neural Network Approximations 2.1 Introduction 2.2 Basics 2.3 Main Results References 3 Generalized Symmetrical Sigmoid Function Induced Banach Space Valued Ordinary and Fractional Neural Network Approximations 3.1 Introduction 3.2 Auxiliary Results 3.3 Main Results References 4 Abstract Multivariate Algebraic Function Induced Neural Network Approximations 4.1 Introduction 4.2 Basic 4.3 Multivariate General Neural Network Approximations References 5 General Multivariate Arctangent Function Induced Neural Network Approximations 5.1 Introduction 5.2 Auxiliary Notions 5.3 Multivariate General Neural Network Approximations References 6 Abstract Multivariate Gudermannian Function Induced Neural Network Approximations 6.1 Introduction 6.2 Background 6.3 Multivariate General Neural Network Approximations References 7 Generalized Symmetrical Sigmoid Function Induced Neural Network Multivariate Approximation 7.1 Introduction 7.2 Auxiliary Results (See Also ch77.14) 7.3 Multivariate General Neural Network Approximations References 8 Quantitative Approximation by Kantorovich-Choquet Quasi-Interpolation Neural Network Operators Revisited 8.1 Introduction 8.2 Background 8.2.1 About the Arctangent Activation Function 8.2.2 About the Algebraic Activation Function 8.2.3 About the Gudermannian Activation Function 8.2.4 About the Generalized Symmetrical Activation Function 8.3 Main Results References 9 Quantitative Approximation by Kantorovich-Shilkret Quasi-interpolation Neural Network Operators Revisited 9.1 Introduction 9.2 Background 9.2.1 About the Arctangent Activation Function 9.2.2 About the Algebraic Activation Function 9.2.3 About the Gudermannian Activation Function 9.2.4 About the Generalized Symmetrical Activation Function 9.3 Main Results References 10 Voronouskaya Univariate and Multivariate Asymptotic Expansions for Sigmoid Functions Induced Quasi-interpolation Neural Network Operators Revisited 10.1 Background 10.1.1 About the Arctangent Activation Function 10.1.2 About the Algebraic Activation Function 10.1.3 About the Gudermannian Activation Function 10.1.4 About the Generalized Symmetrical Activation Function 10.2 Main Results References 11 Univariate Fuzzy Fractional Various Sigmoid Function Activated Neural Network Approximations Revisited 11.1 Introduction 11.2 Fuzzy Fractional Mathematical Analysis Basics 11.3 Real Neural Network Approximation 11.3.1 About the Arctangent Activation Function Neural Networks 11.3.2 About the Algebraic Activation Function Neural Networks 11.3.3 About the Gudermannian Activation Function Neural Networks 11.3.4 About the Generalized Symmetrical Activation Function Neural Networks 11.4 Main Results: Approximation by Fuzzy Quasi-interpolation Neural … References 12 Multivariate Fuzzy Approximation by Neural Network Operators Induced by Several Sigmoid Functions Revisited 12.1 Introduction 12.2 Fuzzy Real Analysis Background 12.3 About Neural Networks Background 12.3.1 About the Arctangent Activation Function 12.3.2 About the Algebraic Activation Function 12.3.3 About the Gudermannian Activation Function 12.3.4 About the Generalized Symmetrical Activation Function 12.4 Main Results: Fuzzy Multivariate Neural Network Approximation Based … References 13 Multivariate Fuzzy-Random and Stochastic Various Activation Functions Activated Neural Network Approximations 13.1 Fuzzy-Random Functions and Stochastic Processes Background 13.2 About Neural Networks Background 13.2.1 About the Arctangent Activation Function 13.2.2 About the Algebraic Activation Function 13.2.3 About the Gudermannian Activation Function 13.2.4 About the Generalized Symmetrical Activation Function 13.3 Main Results References Appendix Conclusion