2000
DOI: 10.1109/72.870037
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Neural-network methods for boundary value problems with irregular boundaries

Abstract: Abstract-Partial differential equations (PDEs) with boundary conditions (Dirichlet or Neumann) defined on boundaries with simple geometry have been successfully treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of complex boundary geometry, where the boundary is determined by a number of points that belong to it and are closely located, so as to offer a reasonable representation. Two networks are employed: a multilayer perceptron and a radial basis function netw… Show more

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Cited by 461 publications
(260 citation statements)
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“…Artificial neural networks based on Broyden-Fletcher-Goldfarb-Shanno (BF GS) optimization technique for solving ordinary and partial differential equations have been excellently presented by Lagaris et al [27]. Furthermore, Lagaris et al [28] investigated neural network methods for boundary value problems with irregular boundaries. Parisi et al [29] presented unsupervised feed forward neural networks for the solution of differential equations.…”
Section: Literature Reviewmentioning
confidence: 99%
“…Artificial neural networks based on Broyden-Fletcher-Goldfarb-Shanno (BF GS) optimization technique for solving ordinary and partial differential equations have been excellently presented by Lagaris et al [27]. Furthermore, Lagaris et al [28] investigated neural network methods for boundary value problems with irregular boundaries. Parisi et al [29] presented unsupervised feed forward neural networks for the solution of differential equations.…”
Section: Literature Reviewmentioning
confidence: 99%
“…The minimization of the networks energy function provides the solution to the system of equations [4]. Lagaris et al [5] employed two networks: a multilayer perceptron and a radial basis function network to solve partial differential equations (PDE) with boundary conditions (Dirichlet or Neumann) defined on boundaries with the case of complex boundary geometry. Mc Fall and Mahan [6] compared weight reuse for two existing methods of defining the network error function; weight reuse is shown to accelerate training of ODE; the second method outperforms the fails unpredictably when weight reuse is applied to accelerate solution of the diffusion equation.…”
Section: Related Workmentioning
confidence: 99%
“…Other non-polynomial splines have been developed in the past, for instance we mention the "Tension Splines" that are based on the exponential function [2]. Neural Networks are well known for their universal approximation capabilities [3], [4] and have been employed for interpolation, approximation and modeling tasks in many cases, ranging from pattern recognition [5], signal processing, control and the solution of ordinary and partial differential equations [6], [7], [8].…”
Section: Rationale and Motivationmentioning
confidence: 99%