Research On Solving Partial Differential Equation Based On Deep Learning

With the development of artificial intelligence and deep learning,the application of artificial intelligence is not limited to traditional tasks such as computer vision,speech and natural language processing.AI for Science has become an important development direction of artificial intelligence,and solving partial differential equations by deep learning is one of the important directions,which has a wide range of application scenarios,such as oil exploration and aerodynamic research,etc.In traditional mathematical methods to solve partial differential equation problems,often need mesh to solve PDE by iteration,which requires high computing cost and storage cost.As for PDE in the arbitrary shape of the region often need higher costs to get accurate solution.And once when the shape of the region changes,the traditional mathematical method needs to solve the PDE start from the begin.Artificial intelligence solves the above problems by introducing operator neural network,which can learn the operator mapping from boundary shape to solution function,and solve the problem of solving repeatedly when the shape of region changing.However,there are still many problems in the process of solving practical problems,such as large demand for data and high cost of actual data collection or data acquisition by traditional methods.This thesis introduces deep operator neural network(Deep ONet)and adopts the idea of physics-informed neural network(PINN)to transform Deep ONet from data driven to physical information constraint driven,so that the neural network no longer needs the numerical solution of each sampling point as the constraint during training.At the same time,it is found that in the modeling of specific problems,because most of the traditional problems are considered in the regular region,it is easy to forget the consistency condition to be met in the irregular region.This thesis adds the consistency condition in the irregular region into the loss function.Secondly,in practical applications,it is found that the physical information under different boundary conditions has a large gap in the order of magnitude in the loss function,which causes great difficulties for the training of neural networks.In this thesis,a dynamic weight optimization algorithm is proposed to make the loss function converge to a better solution.Then,in terms of data representation,traditional Deep ONet uses discrete points to represent the boundary shape.In this thesis,spectral expansion of the boundary is adopted to reduce the dimension required for data representation and make boundary fitting more accurate.Finally,this thesis proposes two processing methods for the solving process of arbitrary shape region: one is to embed the solved region directly into the rectangular region for solving;the other is to map the solved region to the standard rectangular region for solving through nonlinear mapping transformation.The second method is more convenient for data sampling.In order to demonstrate the practicability of the method presented in this thesis,Poisson equation of complex geometric region containing Dirichlet and Neumann boundary conditions,and Robin and Neumann boundary conditions are solved,and an example is given in the hydrogeological application scenario,and the results are compared with those obtained by the traditional method and Comsol.The effectiveness of the proposed method is proved.