Research On Partial Differential Equations Solving Model Based On Physics And CNNs

As an emerging method for solving partial differential equations(PDEs)using deep learning,physics-informed neural networks have attracted a large number of researchers’ attention in recent years,and the researched contents include the partial differential equations solving models and their related applications.In terms of modeling research,convolutional neural networks can effectively learn spatial dependencies due to their parameter sharing property,which makes the physics-informed convolutional neural networks achieve promising results in a series of partial differential equations solving problems.However,when faced with the problem of solving control equations in fluid and heat transfer,the physics-informed convolutional neural networks tends to produce oscillating predictions due to the central difference scheme based on the finite difference method used in the construction of the loss function.In terms of research on model applications,an important scenario is the use of physics-informed neural networks to enhance scientific data based on coarse-grained simulations,i.e.,to recover scientific data with low resolution in time and space into high-resolution data.Existing methods have achieved good recovery results on low-resolution data with small downsampling ratio in time,but the recovery results are not satisfactory when facing time-sparse data in time.The research in this thesis is centered on the partial differential equations solving model based on physical information and convolutional neural networks,which mainly contains the study of the loss function and the study of the temporal super-resolution model of PDEs.The details of the study are as follows:(1)Construction method of loss function for partial differential equations solving model based on finite volume method.In this thesis,we first analyze the causes of oscillating predictions of existing physics-informed convolutional neural networks,and then propose a construction method of loss function based on the finite volume method.The method uses the upwind difference schemes derived from the finite volume method to discretize the PDEs and constructs the loss function based on the discretized equations.The use of the finite volume method ensures that the discretized equations have conservation properties,and the upwind difference scheme ensures that the discretized equations do not introduce downstream information when convection is strong.Through experiments on the steady-state convection-diffusion equations and the Navier-Stokes equations,it is demonstrated that the proposed method effectively mitigates the oscillations of the predicted solutions of the partial differential equation solving model based on physical information and convolutional neural networks,and ensures the prediction accuracy.(2)Transient partial differential equations temporal super-resolution model based on attention mechanism.This thesis further study the temporal super-resolution model for transient PDEs.This thesis first analyzes the problems of existing temporal super-resolution models for scientific data:poor recovery for time-sparse data and the need for high-resolution data for training.To address these problems,this thesis proposes a temporal unit based on the attention mechanism combined with an encoder-decoder structure for predicting the solution of PDEs at intermediate moments.The loss function of the model is modified so that only low-resolution data are required to construct the prediction loss,and the construction method of loss function proposed in study(1)is incorporated.Through experiments on the transient Burgers equations and the Navier-Stokes equations,it is demonstrated that the proposed model has higher recovery accuracy for time-sparse data.