Research On Numerical Solution Of Partial Differential Equations Based On Deep Neural Network

In the process of solving the numerical solution of partial differential equations,neural network is a relatively novel method.This method can directly obtain the experimental solution of the partial differential equation through the input of the point set to approximate the true solution of the equation and meet certain accuracy requirements.In the 1990 s,the theory of neural networks began to develop,limited by the computing power of the computer and could not be widely applied to the field of numerical solution of partial differential equations.In recent years,under the condition of high-performance advanced computers,neural network technology has developed rapidly,and it has begun to adapt to the requirements of computing accuracy and speed in the field of PDEs.Neural networks have good self-learning and self-adaptive capabilities,and the use of deep neural networks to solve numerical solutions of partial differential equations has become a popular application recently.Because the general neural network model method for solving numerical solutions of partial differential equations is difficult to make breakthroughs in solving accuracy,and sometimes cannot meet the needs of practical problems.So we began to seek better model methods to further increase the accuracy of the solution.In order to solve the problem that the general neural network cannot further improve the accuracy of the numerical solution of the partial differential equations,a new adaptive neural network model is proposed to train the multi-step neural network by multi-sampling points within the region.Finally,the numerical solution of the point set on the inner region and the boundary is obtained.The model method can effectively solve the numerical solution of partial differential equations,and the accuracy is higher than that of the general neural network model.It has certain value in the field of numerical solutions of partial differential equations.