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.

N1 - Funding Information:
We thank M. Maniatis, T. Pawlowski and P. Sundell for valuable comments on this work. S. Ossandón acknowledges support from the European Union ’s Horizon 2020, research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 644202 : Geophysical Exploration using Advanced Galerkin Methods (GEAGAM); C. Reyes thanks the partial support from Project Innova-Chile CORFO No. 10CEII-9007: “CSIRO-CHILE International Centre of Excellence in Mining and Mineral Processing”, Program 3-Project 1; and C.M. Reyes acknowledges support from Grant Fondecyt No. 1140781 , DIUBB No. 141709 4/R and the group of Fisica de Altas Energias of the Universidad del Bio-Bio.
Publisher Copyright:
© 2016 Elsevier Ltd

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=84978804167&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2016.06.037

DO - 10.1016/j.camwa.2016.06.037

M3 - Article

AN - SCOPUS:84978804167

VL - 72

SP - 1153

EP - 1163

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 4

ER -