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Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine

In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the...

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Detalles Bibliográficos
Autores principales: Lu, Yanfei, Zhang, Shiqing, Weng, Futian, Sun, Hongli
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Frontiers Media S.A. 2023
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10033520/
https://www.ncbi.nlm.nih.gov/pubmed/36968294
http://dx.doi.org/10.3389/fncom.2023.1120516
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author Lu, Yanfei
Zhang, Shiqing
Weng, Futian
Sun, Hongli
author_facet Lu, Yanfei
Zhang, Shiqing
Weng, Futian
Sun, Hongli
author_sort Lu, Yanfei
collection PubMed
description In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra–Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability.
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spelling pubmed-100335202023-03-24 Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine Lu, Yanfei Zhang, Shiqing Weng, Futian Sun, Hongli Front Comput Neurosci Neuroscience In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra–Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability. Frontiers Media S.A. 2023-03-09 /pmc/articles/PMC10033520/ /pubmed/36968294 http://dx.doi.org/10.3389/fncom.2023.1120516 Text en Copyright © 2023 Lu, Zhang, Weng and Sun. https://creativecommons.org/licenses/by/4.0/This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
spellingShingle Neuroscience
Lu, Yanfei
Zhang, Shiqing
Weng, Futian
Sun, Hongli
Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine
title Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine
title_full Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine
title_fullStr Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine
title_full_unstemmed Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine
title_short Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine
title_sort approximate solutions to several classes of volterra and fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine
topic Neuroscience
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10033520/
https://www.ncbi.nlm.nih.gov/pubmed/36968294
http://dx.doi.org/10.3389/fncom.2023.1120516
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