Application of Neural Networks for Solving Nonlinear Boundary Problems for Complex-Shaped Domains

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.