Multivariate numerical approximation using constructive $$ L^{2} (\mathbb{R}) $$ RBF neural network

Hou, Muzhou; Han, Xuli

doi:10.1007/s00521-011-0604-8

Cited by 14 publications

(2 citation statements)

References 13 publications

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“…The neural network has excellent application potential in many fields (Habib and Qureshi, 2022 ; Li and Ying, 2022 ) owing to its universal function approximation capabilities (Hou and Han, 2012 ; Hou et al, 2017 , 2018 ). In this case, the neural network is widely used as an effective tool for solving differential equations, integral equations, and integro–differential equations (Mall and Chakraverty, 2014 , 2016 ; Jafarian et al, 2017 ; Pakdaman et al, 2017 ; Zuniga-Aguilar et al, 2017 ; Rostami and Jafarian, 2018 ).…”

Section: Introductionmentioning

confidence: 99%

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

Zhang²,

Weng³

et al. 2023

Front. Comput. Neurosci.

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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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Section: Introductionmentioning

confidence: 99%

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

Zhang²,

Weng³

et al. 2023

Front. Comput. Neurosci.

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“…Standard RBF definitions, mostly used in practice, contain Euclid's L2 norm squared as an argument, in order to ensure positive definiteness connected with the invertibility of the interpolation matrix. Their applicability and computational properties have been investigated in theory by many authors, such as in [4] to [9], with belonging calculation improvements investigated in [10] for constructive L2 RBF and machine learning in [11].…”

Section: Introductionmentioning

confidence: 99%

Directional Wind Spectrum Description using Bivariate L1 Norm RBFs

2023

IJEM

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In this paper, the simplest directional wind spectrum description is given using surrogate bivariate polynomial radial basis functions (PRBF) with L1 norm smoothed by dense boundary points distribution, which enables an accurate description of the geometry and the calculation of the volume below the observed surface when belonging double integral is known. For that purpose, the direct solution of double integral below the descriptive surface is given for bivariate polynomial RBFs with integer exponents, which is examined for accuracy on two examples, for Franke's 2D function and upper hemisphere. After proven accurate in those examples, the direct description of the directional wind spectrum and the calculation of the joint density function of the wind spectrum is done in the paper, thus proving PRBFs as an efficient method for wind spectrum description. In that way, it is possible to calculate the joint density function (JDF) of the actual measured directional wind spectrum analytically, instead of the theoretical calculations used so far.

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