Application of Neural Networks to Solve the Dirichlet Problem for Areas of Complex
https://doi.org/10.23947/2587-8999-2024-8-2-68-79
Abstract
Introduction. Many mathematical problems are reduced to solving partial differential equations (PDEs) in domains of complex shapes. Existing analytical and numerical methods do not always provide efficient solutions for such problems. Recently, neural networks have been successfully applied to solve PDEs, typically addressing boundary value problems for domains with simple shapes. This paper attempts to construct a neural network capable of effectively solving boundary value problems for domains of complex shapes.
Materials and Methods. A method for constructing a neural network to solve the Dirichlet problem for regions of complex shape is proposed. Derivatives of singular solutions of the Laplace equation are accepted as activation functions. Singular points of these solutions are distributed along closed curves encompassing the boundary of the domain. The adjustment of the network weights is reduced to minimizing the root-mean-square error during training.
Results. The results of solving Dirichlet problems for various complex-shaped domains are presented. The results are provided in tables, comparing the exact solution and the solution obtained using the neural network. Figures show the domain shapes and the locations of points where the solutions were determined.
Discussion and Conclusion. The presented results indicate a good agreement between the obtained solution and the exact one. It is noted that this method can be easily applied to various boundary value problems. Methods for enhancing the efficiency of such neural networks are suggested.
About the Author
Sh. A. Galaburdin
Don State Technical University
Russian Federation
Russian Federation
Alexander V. Galaburdin, associate professor of the department Mathematics and informatics
1, Gagarin Sq., Rostovon-Don, 344003
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Review
For citations:
Galaburdin Sh.A. Application of Neural Networks to Solve the Dirichlet Problem for Areas of Complex. Computational Mathematics and Information Technologies. 2024;8(2):68-79. https://doi.org/10.23947/2587-8999-2024-8-2-68-79
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