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Volume 21, Issue 5
Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Non-Constant Jumps

Zhiqiang Cai, Junpyo Choi & Min Liu

DOI: 10.4208/ijnam2024-1024

Int. J. Numer. Anal. Mod., 21 (2024), pp. 609-628.

Published online: 2024-10

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  • Abstract

The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in [4, 5]. The method is based on an equivalent least-squares formulation and [5] employs ReLU neural network (NN) functions with ⌈${\rm log}_2(d+1)$⌉$+1$-layer representations for approximating solutions. In this paper, we show theoretically that the method is also capable of accurately approximating non-constant jumps along discontinuous interfaces that are not necessarily straight lines. Theoretical results are confirmed through multiple numerical examples with $d = 2, 3$ and various non-constant jumps and interface shapes, showing that the LSNN method with ⌈${\rm log}_2 (d + 1)$⌉$+1$ layers approximates solutions accurately with degrees of freedom less than that of mesh-based methods and without the common Gibbs phenomena along discontinuous interfaces having non-constant jumps.

  • Keywords

Least-squares method, ReLU neural network, linear advection-reaction equation, discontinuous solution.

  • AMS Subject Headings

65N15, 65N99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-21-609, author = {Cai , ZhiqiangChoi , Junpyo and Liu , Min}, title = {Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Non-Constant Jumps}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2024}, volume = {21}, number = {5}, pages = {609--628}, abstract = {

The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in [4, 5]. The method is based on an equivalent least-squares formulation and [5] employs ReLU neural network (NN) functions with ⌈${\rm log}_2(d+1)$⌉$+1$-layer representations for approximating solutions. In this paper, we show theoretically that the method is also capable of accurately approximating non-constant jumps along discontinuous interfaces that are not necessarily straight lines. Theoretical results are confirmed through multiple numerical examples with $d = 2, 3$ and various non-constant jumps and interface shapes, showing that the LSNN method with ⌈${\rm log}_2 (d + 1)$⌉$+1$ layers approximates solutions accurately with degrees of freedom less than that of mesh-based methods and without the common Gibbs phenomena along discontinuous interfaces having non-constant jumps.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2024-1024}, url = {http://global-sci.org/intro/article_detail/ijnam/23445.html} }
TY - JOUR T1 - Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Non-Constant Jumps AU - Cai , Zhiqiang AU - Choi , Junpyo AU - Liu , Min JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 609 EP - 628 PY - 2024 DA - 2024/10 SN - 21 DO - http://doi.org/10.4208/ijnam2024-1024 UR - https://global-sci.org/intro/article_detail/ijnam/23445.html KW - Least-squares method, ReLU neural network, linear advection-reaction equation, discontinuous solution. AB -

The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in [4, 5]. The method is based on an equivalent least-squares formulation and [5] employs ReLU neural network (NN) functions with ⌈${\rm log}_2(d+1)$⌉$+1$-layer representations for approximating solutions. In this paper, we show theoretically that the method is also capable of accurately approximating non-constant jumps along discontinuous interfaces that are not necessarily straight lines. Theoretical results are confirmed through multiple numerical examples with $d = 2, 3$ and various non-constant jumps and interface shapes, showing that the LSNN method with ⌈${\rm log}_2 (d + 1)$⌉$+1$ layers approximates solutions accurately with degrees of freedom less than that of mesh-based methods and without the common Gibbs phenomena along discontinuous interfaces having non-constant jumps.

Cai , ZhiqiangChoi , Junpyo and Liu , Min. (2024). Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Non-Constant Jumps. International Journal of Numerical Analysis and Modeling. 21 (5). 609-628. doi:10.4208/ijnam2024-1024
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