基于卷积神经网络模型数值求解双曲型偏微分方程的研究  

Study on Numerical Solutions to Hyperbolic Partial Differential Equations Based on the Convolutional Neural Network Model

作　　者：高普阳 赵子桐 杨扬[2] GAO Puyang;ZHAO Zitong;YANG Yang(School of Sciences,Chang’an University,Xi’an 710064,P.R.China;School of Aeronautics,Northwestern Polytechnical University,Xi’an 710072,P.R.China)

机构地区：[1]长安大学理学院,西安710064 [2]西北工业大学航空学院,西安710072

出　　处：《应用数学和力学》2021年第9期932-947,共16页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11901051,11971075);陕西省自然科学基础研究计划青年项目(2020JQ-338,2019JQ-625);长安大学中央高校基本科研业务费(300102120302)。

摘　　要：人工神经网络近年来得到了快速发展,将此方法应用于数值求解偏微分方程是学者们关注的热点问题.相比于传统方法其具有应用范围广泛(即同一种模型可用于求解多种类型方程)、网格剖分条件要求低等优势,并且能够利用训练好的模型直接计算区域中任意点的数值.该文基于卷积神经网络模型,对传统有限体积法格式中的权重系数进行优化,以得到在粗粒度网格下具有较高精度的新数值格式,从而更适用于复杂问题的求解.该网络模型可以准确、有效地求解Burgers方程和level set方程,数值结果稳定,且具有较高数值精度.In recent years,artificial neural networks developed rapidly.Application of this method to partial differential equations became a new idea for exploring numerical solutions to differential equations.Compared with the traditional methods,it has some advantages,such as a wide range of applications(i.e.the same model can be used to solve multiple types of equations)and low meshing requirements.In addition,the trained model can be directly used to calculate the numerical solution at any point in the computation domain.The weight coefficients in the traditional finite volume method were optimized based on the convolutional neural network model to get a new numerical scheme with high-resolution results on the coarse grid.The proposed model helps solve the Burgers and level set equations efficiently and stably with high accuracy.

关 键 词：卷积神经网络模型 BURGERS方程 level set方程 有限体积法 

分 类 号：O29[理学—应用数学]