Application of Neural Networks for Solving Nonlinear Boundary Problems for Complex-Shaped Domains
https://doi.org/10.23947/2587-8999-2024-8-4-35-42
Abstract
Introduction. Many practically significant tasks reduce to nonlinear differential equations. In this study, one of the applications of neural networks for solving specific nonlinear boundary problems for complex-shaped domains is considered. Specifically, the focus is on solving a stationary heat conduction differential equation with a thermal conductivity coefficient dependent on temperature.
Materials and Methods. The original nonlinear boundary problem is linearized through Kirchhoff transformation. A neural network is constructed to solve the resulting linear boundary problem. In this context, derivatives of singular solutions to the Laplace equation are used as activation functions, and these singular points are distributed along closed curves encompassing the boundary of the domain. The weights of the network were tuned by minimizing the mean squared error of training.
Results. Results for the heat conduction problem are obtained for various complex-shaped domains and different forms of dependence of the thermal conductivity coefficient on temperature. The results are presented in tables that contain the exact solution and the solution obtained using the neural network.
Discussion and Conclusion. Based on the computational results, it can be concluded that the proposed method is sufficiently effective for solving the specified type of boundary problems. The use of derivatives of singular solutions to the Laplace equation as activation functions appears to be a promising approach.
Supporting agencies: The author expresses gratitude to the editorial staff for their remarks on the formatting, which allowed the article to be made more accessible
About the Author
A. V. Galaburdin
Don State Technical University
Russian Federation
Russian Federation
Alexander V. Galaburdin - Cand. Sci. (Phys. – math.), Associate Professor at the Department Mathematics and informatics
1, Gagarin Sq., Rostovon-Don, 344003
References
1. Kolmogorov A.N. On the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition. Reports of the USSR Academy of Sciences. 1957;114(5):953–956. (In Russ.)
2. Varshavchik E.A., Galyautdinova A.R., Sedova Yu.S., Tarkhov D.A. Solution of partial differential equations for domains with constant boundaries. Artificial intelligence in solving current social and economic problems of the 21st century: collection of articles. Art. based on materials from the Third All-Russian. scientific-practical conf. Perm, May 14–18, 2018. Perm. state national research univ.; 2018. P. 294. (In Russ.)
3. Epifanov A.A. Application of deep learning methods for solving partial differential equations. Advances in cybernetics. 2020;1(4):22–28. (In Russ.) https://doi.org/10.51790/2712-9942-2020-1-4-3
4. Cai S., Wang Z., Wang S., Perdikaris P., Karniadakis G.E. Physics-informed neural networks for heat transfer problems. Journal of Heat Transfer. 2021;143(6):060801. https://doi.org/10.1115/1.4050542
5. Korsunov N.I., Lomakin A.V. Modeling processes described by a wave differential equation using cellular neural networks. Scientific bulletins. Series History. Political science. Economy. Informatics. 2014;15(186):103–107. (In Russ.)
6. Raissia M., Perdikaris P., Karniadakis G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics. 2019;378:686–707.
7. Zrelova D.P., Ulyanov S.V. Models of physically informed / aware classical Lagrangian / Hamiltonian neural networks in deep learning. Modern information technologies and IT education. 2022;18(2):310–325. (In Russ.) https://doi.org/10.25559/SITITO.18.202202.310-325
8. Shevkun S.A., Samoilov N.S. Application of a neural network approach to solving direct and inverse scattering problems. Engineering School Bulletin FEFU. 2021;2(47):210–225. (In Russ.)
9. Gorbachenko V.I., Zhukov M.V. Solving boundary value problems of mathematical physics using radial basis networks. Journal of Computational Mathematics and Mathematical Physics. 2017;57(1):133–143. (In Russ.)
10. Kuznetsov K.S., Amosova E.V. Numerical solution of the system of Navier-Stokes equations in the case of a compressible medium using neural networks. Bulletin of Tomsk State University. Management, computer technology and information science. 2024;67:31‒41. (In Russ.) https://doi.org/10.17223/19988605/67/4
11. Almajid M. Abu-Alsaud M. Prediction of porous media fluid flow using physics informed neural networks. Journal of Petroleum Science and Engineering. 2021;208:109205. https://doi.org/10.1016/j.petrol.2021.109205
12. Eivazi H.,Tahani M., Schlatter P., Vinuesa R. Physics-informed neural networks for solving Reynolds-averaged Navier-Stokes equations. Physics of Fluids. 2022;34(7):075117. https://doi.org/10.1063/5.0095270
13. Kansa E.J. Motivation for using radial basis functions to solve PDEs. URL: http://uahtitan.uah.edu/kansaweb.html (accessed: 28.11.2024).
14. Kansa E.J. Multiquadrics . A scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 1990;19(8):47–161.
15. Galaburdin A.V. Application of neural networks to solve the Dirichlet problem for areas of complex shape. Computational Mathematics and Information Technologies. 2024;8(2) :68‒79. https://doi.org/10.23947/2587-8999-2024-8-2-68-79 (In Russ.)
16. Brebbia K., Telles J., Vroubel L. Boundary element methods. Moscow: Mir; 1987. 524 p. (In Russ.)
17. Аккuranov Yu.N., Mikhailov V.N. The method of boundary integral equations for solving nonlinear heat transmission problem. USSR Comput. Maths. Math. Phys.1980;20:117‒125
Review
For citations:
Galaburdin A.V. Application of Neural Networks for Solving Nonlinear Boundary Problems for Complex-Shaped Domains. Computational Mathematics and Information Technologies. 2024;8(4):35-42. https://doi.org/10.23947/2587-8999-2024-8-4-35-42
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