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

TL;DR

The LSNN method with ReLU neural network functions 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.

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.

Figures (8)

• Figure 1: A domain decomposition for the case that $\bm\beta$ is piecewise constant
• Figure 1: Approximation results of the problem in \ref{['test1']}
• Figure 1: An illustration of the convergence analysis on one subdomain $\Upsilon_i$
• Figure 2: Approximation results of the problem in \ref{['test2']}
• Figure 3: Approximation results of the problem in \ref{['test3']}

Theorems & Definitions (13)

• Proposition 2.1: see arora2016understandingcai2023least
• Proposition 2.2
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Proof 1
• Lemma 3.4
• Proof 2
• Lemma 5.1
• Proof 3