Artificial Neural Networks and Type-2 Fuzzy Set

Elements of Soft Computing and Its Applications

1st Edition - February 19, 2025
Authors: Snehashish Chakraverty, Arup Kumar Sahoo, Dhabaleswar Mohapatra
Language: English
Paperback ISBN:
9 7 8 - 0 - 4 4 3 - 3 2 8 9 4 - 7
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 3 2 8 9 5 - 4

Soft computing is an emerging discipline which aims to exploit tolerance for imprecision, approximate reasoning, and uncertainty to achieve robustness, tractability, and cost… Read more

Artificial Neural Networks and Type-2 Fuzzy Set

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Soft computing is an emerging discipline which aims to exploit tolerance for imprecision, approximate reasoning, and uncertainty to achieve robustness, tractability, and cost effectiveness for building intelligent machines. Soft computing methodologies include neural networks, fuzzy sets, genetic algorithms, Bayesian networks, and rough sets, among others. In this regard, neural networks are widely used for modeling dynamic solvers, classification of data, and prediction of solutions, whereas fuzzy sets provide a natural framework for dealing with uncertainty. Artificial Neural Networks and Type-2 Fuzzy Set: Elements of Soft Computing and Its Applications covers the fundamental concepts and the latest research on variants of Artificial Neural Networks (ANN), including scientific machine learning and Type-2 Fuzzy Set (T2FS). In addition, the book also covers different applications for solving real-world problems along with various examples and case studies. It may be noted that quite a bit of research has been done on ANN and Fuzzy Set theory/ Fuzzy logic. However, Artificial Neural Networks and Type-2 Fuzzy Set is the first book to cover the use of ANN and fuzzy set theory with regards to Type-2 Fuzzy Set in static and dynamic problems in one place. Artificial Neural Networks and Type-2 Fuzzy Sets are two of the most widely used computational intelligence techniques for solving complex problems in various domains. Both ANN and T2FS have unique characteristics that make them suitable for different types of problems. This book provides the reader with in-depth understanding of how to apply these computational intelligence techniques in various fields of science and engineering in general and static and dynamic problems in particular. Further, for validation purposes of the ANN and fuzzy models, the obtained solutions of each model in the book is compared with already existing solutions that have been obtained with numerical or analytical methods.

Title of Book
Cover image
Title page
Table of Contents
Copyright
Dedication
About the Authors
Preface
Acknowledgements
1: Introduction to soft computing
1.1. Introduction
1.2. Evolution of soft computing
1.3. Soft computing
1.3.1. Neural networks
1.3.2. Fuzzy set theory
1.3.3. Probabilistic reasoning
1.3.4. Evolutionary computation
Part 1: Artificial neural network
Introduction
2: Artificial neural network: an overview
2.1. Introduction to machine learning
2.2. Types of data
2.2.1. Data visualization
2.3. Training of ML model
2.3.1. Hold-out sample validation
2.3.2. K-fold cross-validation
2.4. The birth and evolution of artificial neural network
2.4.1. Topology of ANN
2.4.2. Types of learning
2.4.3. Applications
2.5. Learning rules
2.5.1. Backpropagation learning algorithm
2.6. Configuration in neural network
2.7. Activation functions
2.8. Regularization
2.9. Optimization
2.9.1. Gradient decent
2.9.2. Adam
2.9.3. L-BFGS
3: Mathematical formulation of the neural network for differential equations
3.1. Introduction
3.2. Differential equations
3.3. Incorporating neural networks into ordinary differential equations
3.3.1. Formulation of first-order differential equations with initial conditions
3.3.2. Formulation of second-order differential equations with initial conditions
3.3.3. Formulation of second-order differential equations with boundary conditions
3.4. Example illustration
4: Recent trends in activation functions for solving differential equations
4.1. Introduction
4.2. Methodology
4.3. Simulation results and discussions
4.3.1. Experimental setup
4.3.2. Model problems
4.4. Validation
5: Curriculum learning for an artificial neural network
5.1. Introduction
5.2. Curriculum Learning
5.3. Methodology
5.4. Problem statements
5.4.1. Lane–Emden equation
5.5. Simulation results and discussions
5.5.1. Experimental setup
5.5.2. Model problems
5.6. Validation
6: Symplectic artificial neural network
6.1. Introduction
6.2. Symplectic artificial neural networks
6.3. Methodology
6.3.1. Modeling of symplectic artificial neural network
6.4. Problem statements
6.4.1. Duffing oscillator model
6.4.2. Van der Pol–Mathieu–Duffing oscillator model
6.5. Simulation results and discussions
6.5.1. Experimental setup
6.5.2. Model problems
6.6. Validation
7: Wavelet neural network
7.1. Introduction
7.2. Wavelet neural networks
7.2.1. Mexican Hat wavelet neural networks (MhWNN)
7.3. Methodology
7.3.1. Modeling of Mexican Hat wavelet neural networks
7.4. Problem statement
7.4.1. Non-linear financial model
7.5. Simulation results and discussions
7.5.1. Experimental setup
7.5.2. Model problems
7.6. Validation
8: Physics-informed Neural Network
8.1. Introduction
8.2. Physics-informed Neural Networks
8.3. Methodology
8.4. Problem statement
8.4.1. Modified Fornberg–Whitham equation
8.5. Simulation results and discussions
8.5.1. Experiment setup
8.6. Model problem
8.7. Validation
Part 2: Type-2 fuzzy uncertainty
Introduction
9: Interval and fuzzy set theory: an overview
9.1. Intervals
9.2. Interval arithmetic
9.3. Fuzzy sets
9.3.1. Fuzzy numbers
9.3.2. Parametric form of fuzzy numbers
9.3.3. Double parametric form of fuzzy numbers
9.3.4. Fuzzy arithmetic
9.3.5. Derivative of fuzzy valued function
9.3.6. Riemann–Liouville fractional integral
9.3.7. Fuzzy Riemann–Liouville fractional integral
9.3.8. Caputo fractional derivative
9.3.9. Caputo fuzzy fractional derivative
10: Preliminaries of type-2 fuzzy sets
10.1. Type-2 fuzzy sets
10.1.1. Vertical slice of a type-2 fuzzy set
10.1.2. α˜-Cut of vertical slice
10.1.3. α˜-Plane of a type-2 fuzzy set
10.1.4. Footprint of uncertainty
10.1.5. Lower membership function and upper membership function of a type-2 fuzzy set
10.1.6. Principle set of a type-2 fuzzy set
10.1.7. α-Cut of an α˜-plane
10.2. Triangularly perfect quasi-type-2 fuzzy numbers
10.2.1. Arithmetic of TPQT2FNs
10.3. Gaussian triangular type-2 fuzzy number (GTT2FN)
10.4. General interval type-2 triangular fuzzy number (GIT2TFN)
10.4.1. Perfect interval type-2 triangular fuzzy number (PIT2TFN)
10.5. Derivative of type-2 fuzzy valued functions
10.6. Riemann–Liouville integral of the type-2 fuzzy valued function
10.7. Caputo type-2 fuzzy fractional derivative
11: Uncertain static engineering problems
11.1. Introduction
11.2. Type-2 fuzzy linear system of equations
11.3. Solution procedure for T2FLSEs and FT2FLSEs
11.4. Sample problems
11.5. Application problems
11.5.1. Rectangular sheet
11.5.2. Beam structure
11.5.3. Truss structure
12: Linear dynamical problems with uncertainty
12.1. Introduction
12.2. Type-2 fuzzy linear eigenvalue problems
12.3. Solution procedure of type-2 fuzzy linear eigenvalue problem
12.3.1. Procedure for T2FSEP
12.3.2. Procedure for T2GEP
12.4. Sample problems
12.5. Application problem
12.5.1. Spring-mass system
13: Non-linear dynamical problems with uncertainty
13.1. Introduction
13.2. Type-2 fuzzy non-linear eigenvalue problems
13.3. Solution procedure of the type-2 fuzzy non-linear eigenvalue problem
13.4. Sample problems
13.5. Application problems
13.5.1. Damped spring-mass system
14: Type-2 fuzzy initial value problems with applications
14.1. Introduction
14.2. Problem discussion and proposed methodology
14.3. Sample problems
14.4. Application problems
14.4.1. Logistic model
14.4.2. Electric circuit
14.4.3. Spring-mass system
15: Type-2 fuzzy fractional differential equations with applications
15.1. Introduction
15.2. Legendre wavelet method
15.2.1. Block-pulse function
15.2.2. Fractional-order integration operational matrix
15.3. Problem description and implementation of the LWM
15.4. Sample problems
15.5. Application problems
15.5.1. Fractional-order R-L circuit
15.5.2. Generalized Bagley–Torvik equation
Index

Snehashish Chakraverty

Snehashish Chakraverty has thirty-one years of experience as a researcher and teacher. Presently, he is working in the Department of Mathematics (Applied Mathematics Group), National Institute of Technology Rourkela, Odisha, as a senior (Higher Administrative Grade) professor. Dr Chakraverty received his PhD in Mathematics from IIT-Roorkee in 1993. Thereafter, he did his post-doctoral research at the Institute of Sound and Vibration Research (ISVR), University of Southampton, UK, and at the Faculty of Engineering and Computer Science, Concordia University, Canada. He was also a visiting professor at Concordia and McGill Universities, Canada, during 1997–1999 and visiting professor at the University of Johannesburg, Johannesburg, South Africa, during 2011–2014. He has authored/co-authored/edited 33 books, published 482 research papers (till date) in journals and conferences. He was the president of the section of mathematical sciences of Indian Science Congress (2015–2016) and was the vice president of Orissa Mathematical Society (2011–2013). Prof. Chakraverty is a recipient of prestigious awards, viz. “Careers360 2nd Faculty Research Award” for the Most Outstanding Researcher in the country in the field of Mathematics, Indian National Science Academy (INSA) nomination under International Collaboration/Bilateral Exchange Program (with the Czech Republic), Platinum Jubilee ISCA Lecture Award (2014), CSIR Young Scientist Award (1997), BOYSCAST Fellow. (DST), UCOST Young Scientist Award (2007, 2008), Golden Jubilee Director’s (CBRI) Award (2001), INSA International Bilateral Exchange Award (2015), Roorkee University Gold Medals (1987, 1988) for first positions in MSc and MPhil (Computer Application). He is in the list of 2% world scientists (2020 to 2024) in the Artificial Intelligence and Image Processing category based on an independent study done by Stanford University scientists.

Affiliations and expertise

HAG Professor, Department of Mathematics, Applied Mathematics Group, National Institute of Technology Rourkela, Rourkela, Odisha, India. Differential Equations (ordinary, partial, and fractional), Numerical Analysis, Computational Methods, Structural Dynamics (FGM, Nano), Fluid Dynamics, Mathematical and Uncertainty Modelling, Soft Computing and Machine Intelligence (Artificial Neural Network, Fuzzy, Interval, and Affine Computations).

Arup Kumar Sahoo

Arup Kumar Sahoo is currently working as an Assistant Professor in the Department of Computer Science and Engineering at Siksha “O” Anusandhan (Deemed to be University), Odisha, India. He has joined as a postdoctoral research fellow at the Autonomous Navigation and Sensor Fusion Lab (ANSFL), The Hatter Department of Marine Technologies, University of Haifa, Israel. Dr. Sahoo holds a PhD from the Department of Mathematics, National Institute of Technology Rourkela, Odisha, India. He earned his MPhil in Mathematics from Utkal University, Bhubaneswar, Odisha, India, and MSc in Mathematics and Computing from Biju Patnaik University of Technology, Rourkela, Odisha, India. Dr Sahoo has authored and co-authored 13 research papers and book chapters published in journals and conferences. In 2023, he received the Best Paper Presenter Award at an IEEE Conference.

Affiliations and expertise

Assistant Professor, Department of Computer Science and Engineering, Siksha “O” Anusandhan (Deemed to be University), Odisha, India. Artificial Intelligence, Scientific Machine Learning, Artificial Neural Networks, Advanced Navigation Algorithm, Autonomous Vehicles and Differential Equations.

Dhabaleswar Mohapatra

Dhabaleswar Mohapatra is currently working as an Assistant Professor in the Department of Mathematics at the Institute of Technical Education and Research, Siksha ‘O’ Anusandhan (Deemed to be University), Odisha, India. He received his PhD from the National Institute of Technology, Rourkela, Odisha, India. To date, he has published 12 research articles in journals and book chapters.

Affiliations and expertise

Assistant Professor, Department of Mathematics, Siksha “O” Anusandhan (Deemed to be University), Odisha, India. Type-2 Fuzzy Uncertainty, Static and Dynamic Problems with Uncertainty, Numerical and Analytical Methods, and Fuzzy Differential and Fractional Differential Equations.

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Artificial Neural Networks and Type-2 Fuzzy Set

Elements of Soft Computing and Its Applications

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Snehashish Chakraverty

Arup Kumar Sahoo

Dhabaleswar Mohapatra