Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Non-Constant Jumps
Year: 2024
Author: Zhiqiang Cai, Junpyo Choi, Min Liu
International Journal of Numerical Analysis and Modeling, Vol. 21 (2024), Iss. 5 : pp. 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 ⌈log2(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 ⌈log2(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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2024-1024
International Journal of Numerical Analysis and Modeling, Vol. 21 (2024), Iss. 5 : pp. 609–628
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Least-squares method ReLU neural network linear advection-reaction equation discontinuous solution.
Author Details
Zhiqiang Cai Email
Junpyo Choi Email
Min Liu Email