Neural Network For Solving Partial Differential Equations

Partial differential equations(PDEs)refer to equations of an unknown function and its partial derivatives,describing the relationship between independent variables,the unknown function,and partial derivatives of unknown function.PDEs are one of the most important mathematical tools to describe the laws of the objective physical world.They have important applications in electromagnetics,thermodynamics,fluid mechanics,quantum mechanics,geometry,etc.The exact solutions of PDEs are dif-ficult to obtain,so it is generally considered to obtain approximate solutions of PDEs.Using neural network to solve PDEs is an emerging method of approximate solving PDEs in recent years.Compared with traditional numerical methods,most neural net-work methods do not require meshing,which saves the huge computational and storage costs caused by meshing.In addition,the neural network method for solving PDEs is simple and universal.Therefore,it has attracted the attention of scientific researchers.This thesis is mainly related to the research on related issues of neural network methods for solving PDEs.To sum up,our main research are three-fold:·This thesis discusses the problem of precision-consistency of solving PDEs using neural network methods.This question mainly discusses the error distribution in the domain of the equation.We use an example to illustrate the precision-inconsistency in the approximate solution of PDEs using neural network methods.In order to alleviate this phenomenon,we proposed the Domain decomposition-Search for singular domains-Predictions(DSP)framework.This thesis introduces the implementation details of the DSP framework in detail,and completes the experiments on a Poisson equation,Helmholtz equation and Eikonal equation re-spectively.The experimental results show that the DSP framework we proposed can well alleviate the precision-inconsistency of solving PDEs using neural net-work methods.·This thesis proposes and designs the Multi-Net strategy.Multi-Net strategy is used to replace the traditional Single-Net strategy.Compared with the Single-Net strategy,the Multi-Net strategy can significantly improve the efficiency of the algorithm.In this thesis,we give the definition of Single-Net strategy and Multi-Net strategy,and through time complexity analysis,it is proved that on a large class of PDEs,when the model complexity is close,the algorithm effi-ciency under the Multi-Net strategy is higher than the algorithm efficiency under the Single-Net strategy.In addition,the algorithm under the Multi-Net strategy can solve fractional PDEs,but the algorithm under the Single-Net strategy can not solve such equations.We complete experiments on the Burgers equation,convection-diffusion equation,Kdv equation,Allen-Cahn equation and four spa-tial fractional PDEs,and compare the accuracy and efficiency of different meth-ods under Single-Net strategy and Multi-Net strategy.The experimental results show that,compared with the neural network methods under the Single-Net strat-egy,the methods under the Multi-Net strategy are more efficient.And the neural network methods under the Multi-Net strategy can solve fractional PDEs.·This thesis proposes a neural network method for solving PDEs that considers temporal and spatial dependence:TD-Net.TD-Net improve the accuracy of solv-ing PDEs by modeling temporal and spatial dependence in the neural network method.It models temporal dependency through time discretization technology,and spatial dependency through convolution operation.This thesis introduces the details of TD-Net,and analyzes its time complexity.We complete exper-iments on the Burgers equation,convection-diffusion equation,Kdv equation,Allen-Cahn equation and four spatial fractional PDEs.The experimental results show that,compared with the state-of-the-art neural network methods for solv-ing PDEs,TD-Net has achieved competitive accuracy and fastest efficiency in experimental equations.Finally,we analyzed the limitations of TD-Net.