Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator

Sebastián Ossandón; Mauricio Barrientos; Camilo Reyes

doi:10.1016/j.crme.2017.11.006

Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator

Sebastián Ossandón, Mauricio Barrientos, Camilo Reyes

INSTITUTE OF MATHEMATICS

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3 Scopus citations

Abstract

A numerical method, based on the design of two artificial neural networks, is presented in order to approximate the viscosity and density features of fluids from the eigenvalues of the Stokes operator. The finite element method is used to solve the direct problem by training a first artificial neural network. A nonlinear map of eigenvalues of the Stokes operator as a function of the viscosity and density of the fluid under study is then obtained. This relationship is later inverted and refined by training a second artificial neural network, solving the aforementioned inverse problem. Numerical examples are presented in order to show the effectiveness and the limitations of this methodology.

Original language	English
Pages (from-to)	39-47
Number of pages	9
Journal	Comptes Rendus - Mecanique
Volume	346
Issue number	1
DOIs	https://doi.org/10.1016/j.crme.2017.11.006
State	Published - Jan 2018

Keywords

Artificial neural network
Eigenvalues of the Stokes operator
Finite element method
Inverse problems
Radial basis function
Viscosity and density coefficients

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10.1016/j.crme.2017.11.006

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title = "Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator",

abstract = "A numerical method, based on the design of two artificial neural networks, is presented in order to approximate the viscosity and density features of fluids from the eigenvalues of the Stokes operator. The finite element method is used to solve the direct problem by training a first artificial neural network. A nonlinear map of eigenvalues of the Stokes operator as a function of the viscosity and density of the fluid under study is then obtained. This relationship is later inverted and refined by training a second artificial neural network, solving the aforementioned inverse problem. Numerical examples are presented in order to show the effectiveness and the limitations of this methodology.",

keywords = "Artificial neural network, Eigenvalues of the Stokes operator, Finite element method, Inverse problems, Radial basis function, Viscosity and density coefficients",

author = "Sebasti{\'a}n Ossand{\'o}n and Mauricio Barrientos and Camilo Reyes",

note = "Publisher Copyright: {\textcopyright} 2017 Acad{\'e}mie des sciences",

year = "2018",

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doi = "10.1016/j.crme.2017.11.006",

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T1 - Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator

AU - Ossandón, Sebastián

AU - Barrientos, Mauricio

AU - Reyes, Camilo

PY - 2018/1

Y1 - 2018/1

N2 - A numerical method, based on the design of two artificial neural networks, is presented in order to approximate the viscosity and density features of fluids from the eigenvalues of the Stokes operator. The finite element method is used to solve the direct problem by training a first artificial neural network. A nonlinear map of eigenvalues of the Stokes operator as a function of the viscosity and density of the fluid under study is then obtained. This relationship is later inverted and refined by training a second artificial neural network, solving the aforementioned inverse problem. Numerical examples are presented in order to show the effectiveness and the limitations of this methodology.

AB - A numerical method, based on the design of two artificial neural networks, is presented in order to approximate the viscosity and density features of fluids from the eigenvalues of the Stokes operator. The finite element method is used to solve the direct problem by training a first artificial neural network. A nonlinear map of eigenvalues of the Stokes operator as a function of the viscosity and density of the fluid under study is then obtained. This relationship is later inverted and refined by training a second artificial neural network, solving the aforementioned inverse problem. Numerical examples are presented in order to show the effectiveness and the limitations of this methodology.

KW - Artificial neural network

KW - Eigenvalues of the Stokes operator

KW - Finite element method

KW - Inverse problems

KW - Radial basis function

KW - Viscosity and density coefficients

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SN - 1631-0721

VL - 346

SP - 39

EP - 47

JO - Comptes Rendus - Mecanique

JF - Comptes Rendus - Mecanique

IS - 1

ER -

Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator

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