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 - <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>