Application of Neural Networks for Solving Elliptic Equations in Domains with Complex Geometries

Introduction. Differential equations are often used in modelling across various fields of science and engineering. Recently, neural networks have been increasingly applied to solve differential equations. This paper proposes an original method for constructing a neural network to solve elliptic differential equations. The method is used for solving boundary value problems in domains with complex geometric shapes. Materials and Methods. A method is proposed for constructing a neural network designed to solve partial differential equations of the elliptic type. By applying a transformation of the unknown function, the original problem is reduced to Laplace’s equation. Thus, nonlinear differential equations were considered. In building the neural network, the activation functions are chosen as derivatives of singular solutions to Laplace’s equation. The singular points of these solutions are distributed along closed curves encompassing the boundary of the domain. During the training process, the weights of the network are adjusted by minimizing the mean squared error. Results. The paper presents the results of solving the first boundary value problem for various domains with complex geometries. The results are shown in tables containing both the exact solutions and the solutions obtained using the neural network. Graphical representations of the exact and the neural network-based solutions are also provided. Discussion and Conclusion. The obtained results demonstrate the effectiveness of the proposed neural network construction method in solving various types of elliptic partial differential equations. The method can also be effectively applied to other types of partial differential equations