Application of Neural Networks to Steady-State Oscillations

Introduction. In recent years, the field of mathematics specializing in the application of artificial neural networks has been rapidly developing. In this work, a new method for constructing a neural network for solving wave differential equations is proposed. This method is particularly effective in solving boundary value problems for domains of complex geometric shapes. Materials and Methods. A method is proposed for constructing a neural network designed to solve the wave equation in a planar domain G bounded by an arbitrary closed curve. It is assumed that the boundary conditions are periodic functions of time t, and the steady-state regime is considered. When constructing the neural network, the activation functions are taken as derivatives of singular solutions of the Helmholtz equation. The singular points of these solutions are uniformly distributed along closed curves surrounding the domain boundary. The training set consists of a set of particular solutions of the Helmholtz equation. Results. Results were obtained for the solution of the first boundary value problem in various domains of complex geometric shape and under different boundary conditions. The results are presented in tables containing both the exact solutions of the problem and the solutions obtained using the neural network. A graphical comparison is also provided between the exact solution and the solution obtained with the constructed neural network. Discussion. The presented computational results demonstrate the efficiency of the proposed method for constructing neural networks that solve boundary value problems of partial differential equations in domains of complex geometry. Conclusion. The further development of the proposed method may be applied to solving boundary value problems for the wave equation in exterior domains. Of particular interest is the application of this method to diffraction problems.