Contents
Preface ix
Acknowledgments xiii
Authors xv
Reviewers xvii
1. Preliminaries of Artificial Neural Network 1
1.1 Introduction 1
1.2 Architecture of ANN 2
1.2.1 Feed-Forward Neural Network 3
1.2.2 Feedback Neural Network 3
1.3 Paradigms of Learning 4
1.3.1 Supervised Learning or Associative Learning 4
1.3.2 Unsupervised or Self-Organization Learning 4
1.4 Learning Rules or Learning Processes 5
1.4.1 Error Back-Propagation Learning Algorithm or Delta Learning Rule 5
1.5 Activation Functions 8
1.5.1 Sigmoid Functions 8
1.5.1.1 Unipolar Sigmoid Function 8
1.5.1.2 Bipolar Sigmoid Function 9
1.5.2 Tangent Hyperbolic Function 9
References 9
2. Preliminaries of Ordinary Differential Equations 11
2.1 Definitions 12
2.1.1 Order and Degree of Des 12
2.1.2 Ordinary Differential Equation 12
2.1.3 Partial Differential Equation 12
2.1.4 Linear and Nonlinear Differential Equations 13
2.1.5 Initial Value Problem 13
2.1.6 Boundary Value Problem 14
References 15
3. Multilayer Artificial Neural Network 17
3.1 Structure of Multilayer ANN Model 18
3.2 Formulations and Learning Algorithm of Multilayer ANN Model 18
3.2.1 General Formulation of ODEs Based on ANN Model 18
3.2.2 Formulation of nth-Order IVPs 20
3.2.2.1 Formulation of First-Order IVPs 21
3.2.2.2 Formulation of Second-Order IVPs 21
3.2.3 Formulation of BVPs 22
3.2.3.1 Formulation of Second-Order BVPs 22
3.2.3.2 Formulation of Fourth-Order BVPs 23
3.2.4 Formulation of a System of First-Order ODEs 24
3.2.5 Computation of Gradient of ODEs for Multilayer ANN Model 25
3.3 First-Order Linear ODEs 27
3.4 Higher-Order ODEs 32
3.5 System of ODEs 34
References 36
4. Regression-Based ANN 37
4.1 Algorithm of RBNN Model 37
4.2 Structure of RBNN Model 39
4.3 Formulation and Learning Algorithm of RBNN Model 39
4.4 Computation of Gradient for RBNN Model 40
4.5 First-Order Linear ODEs 40
4.6 Higher-Order Linear ODEs 50
References 56
5. Single-Layer Functional Link Artificial Neural Network 57
5.1 Single-Layer FLANN Models 58
5.1.1 ChNN Model 58
5.1.1.1 Structure of the ChNN Model 58
5.1.1.2 Formulation of the ChNN Model 59
5.1.1.3 Gradient Computation of the ChNN Model 60
5.1.2 LeNNModel 62
5.1.2.1 Structure of the LeNN Model 62
5.1.2.2 Formulation of the LeNN Model 63
5.1.2.3 Gradient Computation of the LeNN Model 63
5.1.3 HeNN Model 64
5.1.3.1 Architecture of the HeNN Model 64
5.1.3.2 Formulation of the HeNN Model 65
5.1.4 Simple Orthogonal Polynomial-Based Neural Network (SOPNN) Model 66
5.1.4.1 Structure of the SOPNN Model 66
5.1.4.2 Formulation of the SOPNN Model 67
5.1.4.3 Gradient Computation of the SOPNN Model 68
5.2 First-Order Linear ODEs 68
5.3 Higher-Order ODEs 69
5.4 System of ODEs 71
References 74
6. Single-Layer Functional Link Artificial Neural Network with Regression-Based Weights 77
6.1 ChNN Model with Regression-Based Weights 78
6.1.1 Structure of the ChNN Model 78
6.1.2 Formulation and Gradient Computation of the ChNN Model 79
6.2 First-Order Linear ODEs 79
6.3 Higher-Order ODEs 83
References 85
7. Lane-Emden Equations 87
7.1 Multilayer ANN-Based Solution of Lane-Emden Equations 89
7.2 FLANN-Based Solution of Lane-Emden Equations 93
7.2.1 Homogeneous Lane-Emden Equations 94
7.2.2 Nonhomogeneous Lane-Emden Equation 101
References 102
8. Emden-Fowler Equations 105
8.1 Multilayer ANN-Based Solution of Emden-Fowler Equations 106
8.2 FLANN-Based Solution of Emden-Fowler Equations 110
References 113
9. Duffing Oscillator Equations 117
9.1 Governing Equation 117
9.2 Unforced Duffing Oscillator Equations 118
9.3 Forced Duffing Oscillator Equations 123
References 131
10. Van der Pol-Duffing Oscillator Equation 133
10.1 Model Equation 134
10.2 Unforced Van der Pol-Duffing Oscillator Equation 135
10.3 Forced Vander Pol-Duffing Oscillator Equation 135
References 144
Index 147