Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Discontinuity Interface

Zhiqiang Cai, Junpyo Choi, Min Liu

Abstract

We studied the least-squares ReLU neural network (LSNN) method for solving linear advection-reaction equation with discontinuous solution in [Cai, Zhiqiang, Jingshuang Chen, and Min Liu. ``Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation.'' Journal of Computational Physics 443 (2021), 110514]. The method is based on a least-squares formulation and uses a new class of approximating functions: ReLU neural network (NN) functions. A critical and additional component of the LSNN method, differing from other NN-based methods, is the introduction of a properly designed and physics preserved discrete differential operator. In this paper, we study the LSNN method for problems with discontinuity interfaces. First, we show that ReLU NN functions with depth $\lceil \log_2(d+1)\rceil+1$ can approximate any $d$-dimensional step function on a discontinuity interface generated by a vector field as streamlines with any prescribed accuracy. By decomposing the solution into continuous and discontinuous parts, we prove theoretically that discretization error of the LSNN method using ReLU NN functions with depth $\lceil \log_2(d+1)\rceil+1$ is mainly determined by the continuous part of the solution provided that the solution jump is constant. Numerical results for both two- and three-dimensional test problems with various discontinuity interfaces show that the LSNN method with enough layers is accurate and does not exhibit the common Gibbs phenomena along discontinuity interfaces.

Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Discontinuity Interface

Abstract

We studied the least-squares ReLU neural network (LSNN) method for solving linear advection-reaction equation with discontinuous solution in [Cai, Zhiqiang, Jingshuang Chen, and Min Liu. ``Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation.'' Journal of Computational Physics 443 (2021), 110514]. The method is based on a least-squares formulation and uses a new class of approximating functions: ReLU neural network (NN) functions. A critical and additional component of the LSNN method, differing from other NN-based methods, is the introduction of a properly designed and physics preserved discrete differential operator. In this paper, we study the LSNN method for problems with discontinuity interfaces. First, we show that ReLU NN functions with depth can approximate any -dimensional step function on a discontinuity interface generated by a vector field as streamlines with any prescribed accuracy. By decomposing the solution into continuous and discontinuous parts, we prove theoretically that discretization error of the LSNN method using ReLU NN functions with depth is mainly determined by the continuous part of the solution provided that the solution jump is constant. Numerical results for both two- and three-dimensional test problems with various discontinuity interfaces show that the LSNN method with enough layers is accurate and does not exhibit the common Gibbs phenomena along discontinuity interfaces.
Paper Structure (15 sections, 9 theorems, 86 equations, 11 figures, 9 tables)

This paper contains 15 sections, 9 theorems, 86 equations, 11 figures, 9 tables.

Table of Contents

  1. Introduction
  2. ReLU NN functions
  3. The LSNN method
  4. Error estimates
  5. Numerical experiments
  6. Two-dimensional problems
  7. A problem with a 3-line segment interface
  8. A problem with a 4-line segment interface
  9. A problem with a curved interface
  10. A problem with a curved interface and $\hat{u}\neq 0$ in \ref{['decop']}
  11. A problem with a sharp transition layer \newlabel2d test5 section0
  12. Three-dimensional problems
  13. A problem with a 2-plane segment interface
  14. A problem with a cylindrical interface
  15. A problem with a spherical interface

Key Result

Proposition 2.1

\newlabelcpwl=relu0 The set of all CPWL functions $f\colon\mathbb{R}^d\to\mathbb{R}$ is equal to ${\@fontswitch{}{\mathcal{}} M}(\lceil \log_2(d+1)\rceil+1)$, i.e., the set of all ReLU NN functions from $\mathbb{R}^d$ to $\mathbb{R}$ that have representations with depth $\lceil \log_2(d+1)\rceil

Figures (11)

  • Figure 1: The neural network function structure
  • Figure 1: An illustration of \ref{['chi-curve lem']}
  • Figure 1: Approximation results of the problem in \ref{['2d test1 section']}
  • Figure 2: $L^2$ norm approximation results of $p(x,y)$ with $\varepsilon=0.001$
  • Figure 2: Approximation results of the problem in \ref{['2d test2 section']}
  • ...and 6 more figures

Theorems & Definitions (20)

  • Proposition 2.1
  • Proof 1
  • Proposition 2.2
  • Proposition 3.1: see bochev2016leastde2004least
  • Proposition 3.2: see bochev2016leastde2004least
  • Lemma 3.3: see Cai2021linear
  • Remark 3.4
  • Remark 3.5
  • Lemma 4.1
  • Proof 2
  • ...and 10 more