Application And Research Of Neural Network Methods For Allen-Cahn Equation

Allen-cahn equation and its derived equation are widely used in multidirectional incompressible flow,microstructure evolution,crystal growth and other natural phenomena,and the study of its numerical solution algorithm is of great significance.This paper firstly reviews the research status of numerical solution of Allen-Cahn equation and neural network solution of differential equation,introduces the allen-Cahn equation and its characteristics and the basic knowledge of neural network,and expounds the basic principles of the construction solution method of differential equation and PINN neural network method.Secondly,for the one-dimensional Allen-Cahn equation with Dirichlet boundary condition and periodic boundary condition,the neural network structure,numerical solution format,minimum value problem and solution steps of the construction solution method and PINN method are given.Numerical examples are used to verify the validity of the two neural network solutions,but there are some limitations such as the difficulty in constructing corresponding numerical schemes and the failure of PINN method to solve allen-Cahn equations with periodic boundary conditions.Then,the implicit Runge-Kutta method is used to advance in the time direction and the PINN method is used to construct the improved neural network solution based on the PINN method in the space direction.The neural network structure,numerical format,minimum value problem and solving steps of the method are given to solve allen-Cahn equation.The method is used to solve one-dimensional and two-dimensional Allen-Cahn equations with different boundary types.The results show that the improved PINN method is suitable for solving all kinds of boundary conditions.In addition,numerical examples are used to analyze the influence of multiple hyperparameters in the improved PINN method on the solution results.Finally,under reasonable assumptions,the Allen-Cahn equation model for the growth process of double grains was established,and the grain growth was simulated based on the improved PINN method.The simulation results show that the spherical grains shrink and the volume fraction decreases linearly with time,which is consistent with the results obtained by the difference method and theoretical analysis.