Neural network approach for solving nonlocal boundary value problems

This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving...

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Vydáno v:Neural computing & applications Ročník 32; číslo 17; s. 14153 - 14171
Hlavní autoři: Palade, V., Petrov, M. S., Todorov, T. D.
Médium: Journal Article
Jazyk:angličtina
Vydáno: London Springer London 01.09.2020
Springer Nature B.V
Témata:
Artificial Intelligence
Boundary conditions
Boundary value problems
Computational Biology/Bioinformatics
Computational Science and Engineering
Computer Science
Data Mining and Knowledge Discovery
Differential equations
Dirichlet problem
Divergence
Finite element method
Image Processing and Computer Vision
Learning
Mathematical analysis
Methods
Neural networks
Numerical integration
Original Article
Probability and Statistics in Computer Science
Radial basis function
Variational methods
RBF neural network
Nonlocal nonlinear problem
Two-point step size gradient method
Variable learning rate
ISSN:0941-0643, 1433-3058
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Shrnutí:This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving boundary value problems by using neural networks. A numerical integration procedure is developed for computing the local L 2 -inner product. It is known that the non-variational methods are not effective in solving nonlocal problems. In this paper, the weak formulation of the nonlocal problem is reduced to the minimization of a nonlinear functional. Unlike many previous works, we use an integral objective functional for implementing the learning procedure. Well-distributed nodes are used as the centers of the RBF neural network. The weights of the RBF network are determined by a two-point step size gradient method. The neural network method proposed in this paper is an alternative to the finite-element method (FEM) for solving nonlocal boundary value problems in non-Lipschitz domains. A new variable learning rate strategy has been developed and implemented in order to avoid the divergence of the training process. A comparison between the proposed neural network approach and the FEM is illustrated by challenging examples, and the performance of both methods is thoroughly analyzed.
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ISSN:0941-0643
1433-3058
DOI:10.1007/s00521-020-04810-0