Learning Data for Neural-Network-Based Numerical Solution of PDEs: Application to Dirichlet-to-Neumann Problems

We propose neural-network-based algorithms for the numerical solution of boundary-value problems for the Laplace equation. Such a numerical solution is inherently mesh-free, and in the approximation process, stochastic algorithms are employed. The chief challenge in the solution framework is to gene...

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Bibliographic Details
Published in:Algorithms 2023-02, Vol.16 (2), p.111
Main Authors: Izsák, Ferenc, Djebbar, Taki Eddine
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Boundary value problems
Dirichlet problem
Dirichlet-to-Neumann problem
Electrical impedance
Laplace equation
Learning
learning data
Mathematical models
Meshless methods
method of fundamental solutions
Neumann problem
Neural networks
Numerical analysis
Partial differential equations
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Values
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Summary:We propose neural-network-based algorithms for the numerical solution of boundary-value problems for the Laplace equation. Such a numerical solution is inherently mesh-free, and in the approximation process, stochastic algorithms are employed. The chief challenge in the solution framework is to generate appropriate learning data in the absence of the solution. Our main idea was to use fundamental solutions for this purpose and make a link with the so-called method of fundamental solutions. In this way, beyond the classical boundary-value problems, Dirichlet-to-Neumann operators can also be approximated. This problem was investigated in detail. Moreover, for this complex problem, low-rank approximations were constructed. Such efficient solution algorithms can serve as a basis for computational electrical impedance tomography.
ISSN:1999-4893
1999-4893
DOI:10.3390/a16020111