APPLICATION OF PHYSICALLY-ORIENTED NEURAL NETWORKS TO SOLVING PROBLEMS OF LINEAR PARTIAL DERIVATIVE EQUATIONS WITH INHOMOGENEOUS MIXED BOUNDARY CONDITIONS

Abstract

The paper considers the application of Physics-Informed Neural Networks (PINN) to solving a two-dimensional Poisson equation with inhomogeneous mixed boundary conditions. The PINN method combines neural network approximations and physical laws, including the residual of the equation and errors at the boundary, into a single loss function. The model is implemented in the PyTorch environment and trained on a sample generated by the Latin hypercubic sampling method. To verify the accuracy, a comparison is made with an analytical solution, as well as with traditional numerical methods. The influence of key hyperparameters is systematically studied: the number of training points, weights in the loss function, network architecture, and training step. It is found that increasing the weight on the boundary conditions improves convergence and accuracy, and optimal architectures with 4-5 hidden layers and 64 neurons provide a relative error of less than 0.1-0.2%. Unlike numerical schemes requiring grid discretization, PINN has demonstrated versatility and high accuracy when dealing with sparse data and complex conditions, making it a promising tool for computational mathematics and modeling of physical processes in conditions of complex geometry and incomplete information.