Neural network solution for an inverse problem associated with the Dirichlet eigenvalues of the anisotropic Laplace operator

Sebastián Ossandón, Camilo Reyes, Carlos M. Reyes

Resultado de la investigación: Article

5 Citas (Scopus)

Resumen

An innovative numerical method based on an artificial neural network is presented in order to solve an inverse problem associated with the calculation of the Dirichlet eigenvalues of the anisotropic Laplace operator. Using a set of predefined eigenvalues obtained by solving repeatedly the direct problem, a radial basis neural network is designed with the purpose to find the appropriate components of the anisotropy matrix, related to the Laplace operator, and thus solving the associated inverse problem. The finite element method is used to solve the direct problem and to create the training set for the first radial basis neural network. A nonlinear map of the Dirichlet eigenvalues as a function of the anisotropy matrix is then obtained. This nonlinear relationship is later inverted and refined, by training a second radial basis neural network, solving the aforementioned inverse problem. Some numerical examples are presented to prove the effectiveness of the introduced method.

Idioma originalEnglish
Páginas (desde-hasta)1153-1163
Número de páginas11
PublicaciónComputers and Mathematics with Applications
Volumen72
N.º4
DOI
EstadoPublished - 1 ago 2016
Publicado de forma externa

Huella dactilar

Dirichlet Eigenvalues
Laplace Operator
Inverse problems
Mathematical operators
Inverse Problem
Neural Networks
Neural networks
Anisotropy
Nonlinear Map
Artificial Neural Network
Finite Element Method
Numerical Methods
Eigenvalue
Numerical Examples
Numerical methods
Finite element method
Training

ASJC Scopus subject areas

  • Modelling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

Citar esto

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abstract = "An innovative numerical method based on an artificial neural network is presented in order to solve an inverse problem associated with the calculation of the Dirichlet eigenvalues of the anisotropic Laplace operator. Using a set of predefined eigenvalues obtained by solving repeatedly the direct problem, a radial basis neural network is designed with the purpose to find the appropriate components of the anisotropy matrix, related to the Laplace operator, and thus solving the associated inverse problem. The finite element method is used to solve the direct problem and to create the training set for the first radial basis neural network. A nonlinear map of the Dirichlet eigenvalues as a function of the anisotropy matrix is then obtained. This nonlinear relationship is later inverted and refined, by training a second radial basis neural network, solving the aforementioned inverse problem. Some numerical examples are presented to prove the effectiveness of the introduced method.",
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Neural network solution for an inverse problem associated with the Dirichlet eigenvalues of the anisotropic Laplace operator. / Ossandón, Sebastián; Reyes, Camilo; Reyes, Carlos M.

En: Computers and Mathematics with Applications, Vol. 72, N.º 4, 01.08.2016, p. 1153-1163.

Resultado de la investigación: Article

TY - JOUR

T1 - Neural network solution for an inverse problem associated with the Dirichlet eigenvalues of the anisotropic Laplace operator

AU - Ossandón, Sebastián

AU - Reyes, Camilo

AU - Reyes, Carlos M.

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AB - An innovative numerical method based on an artificial neural network is presented in order to solve an inverse problem associated with the calculation of the Dirichlet eigenvalues of the anisotropic Laplace operator. Using a set of predefined eigenvalues obtained by solving repeatedly the direct problem, a radial basis neural network is designed with the purpose to find the appropriate components of the anisotropy matrix, related to the Laplace operator, and thus solving the associated inverse problem. The finite element method is used to solve the direct problem and to create the training set for the first radial basis neural network. A nonlinear map of the Dirichlet eigenvalues as a function of the anisotropy matrix is then obtained. This nonlinear relationship is later inverted and refined, by training a second radial basis neural network, solving the aforementioned inverse problem. Some numerical examples are presented to prove the effectiveness of the introduced method.

KW - Anisotropic Laplace operator

KW - Artificial neural networks

KW - Dirichlet eigenvalues

KW - Finite element method

KW - Inverse problems

KW - Radial basis functions

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