@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 = {<p style="text-align: justify;">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.</p>},
issn = {2617-8710},
doi = {https://doi.org/10.4208/ijnam2024-1024},
url = {http://global-sci.org/intro/article_detail/ijnam/23445.html}
}