Research On The Solution And Construction Of Differential Equations Under Deep Learning Framework

Since the beginning of the 18th century,people have studied differential equations in combination with physical problems.As an important branch of mathematics,differential equations have an important position in scientific research.Generally speaking,the research on differential equations in the field of mathematics focuses on several different aspects,most scholars mainly focus on the positive and inverse problems of differential equations.The positive problem of differential equations is to focus on the solution of differential equations,that is,the study of solving differential equations.If some of the original conditions in the definite solution of differential equations become unknown conditions,and the unknown function of the original equation may still be unknown,or only some information related to this unknown function is known.We have to determine these unknown quantities through the equation,the solution condition and some additional conditions.This is the differential equation inverse problem research,that is,the research on the structure of differential equations.The application range of the research on the solution and structure of differential equations is very wide,such as medicine,chemical industry,aerospace and other fields.Therefore,the research on the solution and structure of differential equations has very important significance and research value.The thesis first introduces the research on the solution and construction of differential equations,the characteristics of neural networks and the research methods of using neural networks to solve and construct differential equations,etc.Because some positive and inverse problem of equations are difficult to solve,and traditional numerical methods have specific shortcomings.For example,the finite difference method uses orthogonal networks,which are difficult to adapt to the arbitrariness of the regional shape,and the large workload of the finite element method severely limits the practical problems.In recent years,with the rapid development of computing and storage capabilities,new opportunities have been provided for the research of differential equations.Therefore,how to find new methods for solving differential equations and structural research has become an urgent topic for people to solve.Inspired by the design of neural networks in deep learning,the paper proposes two improved neural network algorithms based on Physics Informed deep neural network:differential equation solving algorithm and differential equation construction algorithm to achieve two goals:accuracy predict differential equation dynamics models and reveal hidden partial differential equation models.For predictive differential equation dynamics models,the basic idea of deep neural networks is to approximate the differential equation dynamics system through nonlinear mapping.This method does not need to know the analytical solution form of the differential equation,it is an unsupervised deep learning method.For revealing the hidden partial differential equation model,the basic idea of deep neural network is to approximate the analytical solution of the equation by nonlinear mapping under the prior information of the known analytical solution of the equation.At the same time,it satisfies the partial differential equation dynamic system to obtain the potential partial differential equation model.Compared with the existing methods,the deep neural network algorithm for solving the differential equation proposed in this paper is more flexible,and the differential equation dynamic model predicts the solution of the differential equation is not required as a prior information,and the error is small compared with other learning methods.The differential equation construction algorithm can solve the variable parameter inverse problem of partial differential equations.The two deep learning algorithm solutions constructed can be reused at any time.Numerical value The simulation results show that the method proposed in the paper is feasible and effective,and has good accuracy.