Physics-informed neural networks for modeling steady-state heat conduction

The paper presents the results of research on the application of physics-informed neural networks (PINN) for solving the steady state heat conduction equation. The solutions of the heat conduction equation with boundary conditions of I, II and III kinds, and also taking into account the presence of a heat source are considered. Formulations of heat conduction problems and error functions for one-dimensional and two-dimensional cases are given. The proposed neural network is implemented using PyTorch and DeepXDE frameworks. Based on the import of mesh data from the digital product “Logos Heat”, the possibility of using geometry of arbitrary type is provided. The influence of the neural network architecture and the choice of the activation function on the obtained results is investigated. Numerical experiments for the proposed method are carried out on one-dimensional and two-dimensional problems having exact and known numerical solution.