Electrophysics based on Laplace’s theorem and physical information neural networks

Focusing on the field of electrophysics, this paper provides an in-depth study of mathematical models based on the combination of Laplace's Theorem and Physical Neural Networks (PINN) and their applications. Laplace's theorem, as a key partial differential equation describing steady state physical phenomena, is widely used in steady state field analysis in electrophysics, such as the potential distribution of electrostatic fields in passive regions, etc. PINN is a recent innovative approach, which provides a powerful tool for solving partial differential equations by cleverly embedding the laws of physics into neural networks. In this paper, the basic principles of PINN are firstly explained in detail, including the construction of the network architecture, the loss function, and the way of integrating the physical equations and boundary conditions into the neural network training. Subsequently, the proposed method is explored in depth by elaborating a simple two-dimensional potential problem. In this problem, specific boundary conditions are set to simulate the potential constraints in a real physical scenario. The two-dimensional potential problem is solved by the PINN method to demonstrate that PINN can efficiently solve the steady-state electric field distribution under the condition of satisfying Laplace's theorem. At the same time, the results of PINN are compared and analysed with the traditional numerical methods to verify the accuracy, efficiency and adaptability of PINN under complex boundary conditions, and to provide new ideas and methods for related research and engineering applications in the field of electrophysics.