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arXiv · 2306.07445

Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Non-constant Jumps

Zhiqiang Cai·Junpyo Choi·Min Liu
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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 \cite{Cai2021linear,cai2023least}. The method is based on an equivalent least-squares formulation and \cite{cai2023least} employs ReLU neural network (NN) functions with $\lceil \log_2(d+1)\rceil+1$-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 $\lceil \log_2(d+1)\rceil+1$ 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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Zhiqiang Cai, Junpyo Choi, Min Liu. 2024-05-29. Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Non-constant Jumps. https://doi.org/10.4208/ijnam2024-1024

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