A MESH-LESS, RAY-BASED DEEP NEURAL NETWORK METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH FREQUENCY

被引：0
作者：
Yang, Andy L. [1 ]
Gu, Feng [2 ]
机构：
[1] Dutchfork High Sch, Irmo, SC USA
[2] CUNY Coll Staten Isl, Dept Comp Sci, Staten Isl, NY 10314 USA
来源：
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING | 2022年 / 19卷 / 04期
关键词：
Deep learning; plane wave; deep Neural Network; loss; high frequency; Helmholtz equation; DISCONTINUOUS GALERKIN METHODS; NUMERICAL-SOLUTION; WAVE; APPROXIMATION; ALGORITHM; FEM;
D O I：
暂无
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
This article introduces a mesh-less, ray-based deep neural network method to solve the Helmholtz equation with high frequency. This method does not use an adaptive mesh refinement method, nor does it design a numerical scheme using some specially designed basis function to calculate the numerical solution, but it has the advantages of easy implementation and no mesh. We have carried out various numerical examples to prove the accuracy and efficiency of the proposed numerical method.
引用
收藏
页码：587 / 601
页数：15
相关论文
共 33 条
  • [1] Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?
    Babuska, IM
    Sauter, SA
    [J]. SIAM JOURNAL ON NUMERICAL ANALYSIS, 1997, 34 (06) : 2392 - 2423
  • [2] THE NUMERICAL-SOLUTION OF THE HELMHOLTZ-EQUATION FOR WAVE-PROPAGATION PROBLEMS IN UNDERWATER ACOUSTICS
    BAYLISS, A
    GOLDSTEIN, CI
    TURKEL, E
    [J]. COMPUTERS & MATHEMATICS WITH APPLICATIONS, 1985, 11 (7-8) : 655 - 665
  • [3] Beck C., 2020, ARXIV PREPRINT ARXIV
  • [4] Bouche D., 1997, Asymptotic Methods in Electromagnetics
  • [5] A PHASE SHIFT DEEP NEURAL NETWORK FOR HIGH FREQUENCY APPROXIMATION AND WAVE PROBLEMS
    Cai, Wei
    Li, Xiaoguang
    Liu, Lizuo
    [J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2020, 42 (05) : A3285 - A3312
  • [6] Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem
    Cessenat, O
    Despres, B
    [J]. SIAM JOURNAL ON NUMERICAL ANALYSIS, 1998, 35 (01) : 255 - 299
  • [7] A HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER
    Chen, Huangxin
    Lu, Peipei
    Xu, Xuejun
    [J]. SIAM JOURNAL ON NUMERICAL ANALYSIS, 2013, 51 (04) : 2166 - 2188
  • [8] PLANE WAVE DISCONTINUOUS GALERKIN METHODS: ANALYSIS OF THE h-VERSION
    Gittelson, Claude J.
    Hiptmair, Ralf
    Perugia, Ilaria
    [J]. ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 2009, 43 (02): : 297 - 331
  • [9] Error bounds for approximations with deep ReLU neural networks in Ws,p norms
    Guehring, Ingo
    Kutyniok, Gitta
    Petersen, Philipp
    [J]. ANALYSIS AND APPLICATIONS, 2020, 18 (05) : 803 - 859
  • [10] RELU DEEP NEURAL NETWORKS AND LINEAR FINITE ELEMENTS
    He, Juncai
    Li, Lin
    Xu, Jinchao
    Zheng, Chunyue
    [J]. JOURNAL OF COMPUTATIONAL MATHEMATICS, 2020, 38 (03) : 502 - 527
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