Neural Network Method for Solving Partial Differential Equation

As a powerful information processing tool, neural network has been widely used in the fields of computer vision, biomedicine, and oil and gas engineering, which has triggered a technological revolution in many fields. In recent years, the solution of partial differential equations based on deep learning has become a new research hotspot. Following the traditional terms of analytical solutions and numerical solutions of partial differential equations, this paper calls the method of solving partial differential equations using neural networks as partial differential equation intelligent solution method or partial differential equation neural network solution method. This paper first briefly introduces the development of intelligent solution of partial differential equations, and then discusses from the perspective of inversion of unknown partial differential equations and solution of known partial differential equations, focusing on the solution methods of known partial differential equations. In this paper, we propose a new class of squareplus kernel functions and generalized computer network kernel functions, and compare them with Gaussian kernel functions. In order to verify the effectiveness of the model solution, we separately analyze the differential equations with the first and second boundary conditions, and focus on the impact of the number of center points and different kernel functions on the RBF-PINN model solution error, And the adaptability of the center point and the corresponding kernel width parameters. The experimental results show that the RBF-PINN model has achieved higher accuracy than the PINN model.