基于图卷积的一维平流方程空间离散化数值求解加速方法  

Accelerated Numerical Method for Discretization of One-Dimensional Advection Equations Based on Graph Convolution

作　　者：时津津 宋宁 田浩 聂婕[1] 魏志强[1] Shi Jinjin;Song Ning;Tian Hao;Nie Jie;Wei Zhiqiang(Faculty of Information Science and Engineering,Ocean University of China,Qingdao 266100,China;School of Mathematical Sciences,Ocean University of China,Qingdao 266100,China)

机构地区：[1]中国海洋大学信息科学与工程学部,山东青岛266100 [2]中国海洋大学数学科学学院,山东青岛266100

出　　处：《中国海洋大学学报（自然科学版）》2024年第1期138-143,164,共7页Periodical of Ocean University of China

基　　金：国家重点研究发展计划项目(2020YFB1711700);中央高校基本科研业务费专项资金项目(202042008);国家自然科学基金项目(62172376,62072418);山东省重大科技创新工程项目(2019JZZY020705)资助。

摘　　要：本文提出了一种基于图卷积神经网络的偏微分方程空间离散化数值求解加速方法,并将该方法应用于求解一维平流方程的研究中,实现了一维平流方程的加速求解。并设计了基于图卷积的一维平流方程空间离散化神经网络模型(GCPNN),其在物理先验知识指导下基于图模型利用空间图结构特征进行一维平流方程空间离散化求解加速方案建模,在构建图结构关系过程中,基于物理先验知识建立邻接矩阵,利用邻接矩阵融合了全局信息,从而实现了一维平流方程的加速求解。并且通过设计对比实验和消融实验验证了基于GCPNN的求解器相较于基线求解器和CNN求解器在求解精度和计算成本方面的优势,且验证了加入物理先验知识指导及全局信息融合的有效性。An accelerated numerical solution method of spatial discretization of partial differential equations based on graph convolution neural network is proposed.This method is applied to the study of solving one-dimensional advection equations,realizing the accelerated solution of one-dimensional advection equations.The authors designed a one-dimensional advection equation space discretization neural network model(GCPNN)based on graph convolution.Under the guidance of physical prior knowledge,it modeled the acceleration scheme of one-dimensional advection equation space discretization based on graph model using spatial graph structure characteristics.In the process of building graph structure relationships,adjacency matrix was established based on physical prior knowledge,and global information was fused using adjacency matrix.Thus,the accelerated solution of one-dimensional advection equations was achieved.Through design comparison experiments and ablation experiments,the advantages of GCPNN based solvers in solving accuracy and computational cost compared to baseline solvers and CNN solvers were verified,and the effectiveness of incorporating physical prior knowledge guidance and global information fusion was also verified.

关 键 词：一维平流方程 数值求解加速 图卷积 空间离散化 

分 类 号：O242.1[理学—计算数学]