Numerical solution of elliptic partial differential equation using radial basis function neural networks

Li, Jianyu; Luo, Siwei; Qi, Yingjian; Huang, Y.J.

doi:10.1016/s0893-6080(03)00083-2

Cited by 119 publications

(42 citation statements)

References 15 publications

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“…He restricted his work to second order partial di erential equations. Jianyu et al [187] used neural network for solving di erential equations. In their work a new growing RBF-node insertion strategy is used in order to improve the net performance.…”

Section: Rbfn In DI Erential Equationsmentioning

confidence: 99%

Where are Universals?

Peacock¹

2016

Metaphysica

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Radial basis function networks (RBFNs) have gained widespread appeal amongst researchers and have shown good performance in a variety of application domains. They have potential for hybridization and demonstrate some interesting emergent behaviors. This paper aims to o er a compendious and sensible survey on RBF networks. The advantages they o er, such as fast training and global approximation capability with local responses, are attracting many researchers to use them in diversi ed elds. The overall algorithmic development of RBF networks by giving special focus on their learning methods, novel kernels, and ne tuning of kernel parameters have been discussed. In addition, we have considered the recent research work on optimization of multi-criterions in RBF networks and a range of indicative application areas along with some open source RBFN tools.

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Section: Rbfn In DI Erential Equationsmentioning

confidence: 99%

Where are Universals?

Peacock¹

2016

Metaphysica

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“…To train these networks, weights are adjusted in a manner that difference between the outputs of the network and targets will be minimized. Therefore, in computing the Runge Kutta coefficients by using MLP neural network, input and target data are considered starting point and end point, respectively in each step of integration (v) and the values of analytical solution which are achieved by equations (10), are used as starting point for calculating each step. The Runge Kutta will solve the system of third order equations (14) as follow: Given equations (9), (10) and (14): Therefore, the solution of Runge Kutta for two-body problem will be given by equations (32) are determined by the weights of network.…”

Section: Modeling Runge Kutta Coefficients Calculation By Mlpmentioning

confidence: 99%

“…Research for solving OED and Partial Differential Equations (PDE) by using ANN have been progressed significantly during the last decade. Some of these works are: converting PDE to ODE and solving them as an infinite objective function minimization [4], using Feed Forward Neural Network (FFNN) which can approximate linear and nonlinear OED [5], [6], creating cellular neural network which can approximate the solution of different PDEs [7], using FFNN for solving a special class of the linear first-order PDE [8], using of Radial Basis Function Neural Network (RBFNN) [9] for solving linear differential equations [10], presenting ANN with an embedded finite-element model, for the solution of PDE [11]. In all above mentioned researches, ANN acts as direct solution of differential equations.…”

Section: Introductionmentioning

confidence: 99%

ANN-based modeling of third order runge kutta method

Dehghanpour¹,

Rahati²,

Dehghanian³

2015

JACST

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The world's common rules (Quantum Physics, Electronics, Computational Chemistry and Astronomy) find their normal mathematical explanation in language of differential equations, so finding optimum numerical solution methods for these equations are very important. In this paper, using an artificial neural network (ANN) a numerical approach is designed to solve a specific system of differential equations such that the training process of the ANN calculates the optimal values for the coefficients of third order Runge Kutta method. To validate our approach, we performed some experiments by solving two body problem using coefficients obtained by ANN and also two other well-known coefficients namely Classical and Heun. The results show that the ANN approach has a better performance in compare with two other approaches.

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“…Tsoulos et al [36] solved differential equations with neural networks using a scheme that worked on the basis of grammatical evolution. Numerical solution of elliptic partial differential equation using radial basis function neural networks has been presented by Jianyu et al [37]. Shirvany et al [38] proposed multilayer perceptron and radial basis function (RBF ) neural networks with a new unsupervised training method for numerical solution of partial differential equations.…”

Section: Literature Reviewmentioning

confidence: 99%

Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

Khalid¹,

Sultana²,

Zaidi³

2014

IJCA

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In the current paper, a neural network method to solve sixth-order differential equations and their boundary conditions has been presented. The idea this method incorporates is to integrate knowledge about the differential equation and its boundary conditions into neural networks and the training sets. Neural networks are being used incessantly to solve all kinds of problems hailing a wide range of disciplines. Several examples are given to illustrate the efficiency and implementation of the Neural Network method.

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Numerical solution of elliptic partial differential equation using radial basis function neural networks

Cited by 119 publications

References 15 publications

Where are Universals?

Where are Universals?

ANN-based modeling of third order runge kutta method

Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

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