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Two-scale Neural Networks for Partial Differential Equations with Small Parameters

Qiao Zhuang, Chris Ziyi Yao, Zhongqiang Zhang, George Em Karniadakis

Abstract

We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.

Two-scale Neural Networks for Partial Differential Equations with Small Parameters

Abstract

We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.
Paper Structure (7 sections, 22 equations, 19 figures, 6 tables, 1 algorithm)

This paper contains 7 sections, 22 equations, 19 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Related works
  4. Contribution and significance of the work
  5. Formulation and algorithms
  6. Numerical results
  7. Conclusion and discussion

Figures (19)

  • Figure 1: Numerical results for Example \ref{['exm:1p1']} when $\epsilon=10^{-2}$ using $N(x, (x-0.5)/\sqrt{\epsilon}, 1/\sqrt{\epsilon})$, with parameters specified in Table \ref{['tab:exm1p1_para']} .
  • Figure 2: Numerical results for Example \ref{['exm:1p1']} when $\epsilon=10^{-3}$, using $N(x, (x-0.5)/\sqrt{\epsilon}, 1/\sqrt{\epsilon})$, with parameters specified in Table \ref{['tab:exm1p1_para']} .
  • Figure 3: Numerical results for Example \ref{['exm:1p1']} when $\epsilon=10^{-4}$ ((a) to (d)) and $10^{-5}$ ((e) to (h)), with parameters specified in Table \ref{['tab:exm1p1_para']}.
  • Figure 4: Numerical results for Example \ref{['exm:1p4']} when $\epsilon=10^{-2}$ using $N(x, x/\sqrt{\epsilon}, 1/\sqrt{\epsilon})$, with parameters specified in Table \ref{['tab:exm1p4_para']}.
  • Figure 5: Loss and errors history, as well as the statistical metrics $\overline{e}$ and $\overline{e}\pm \sigma$ for Examples \ref{['exm:1p1']} and \ref{['exm:1p4']} when $\epsilon=10^{-2}$ and $\epsilon=10^{-3}$.
  • ...and 14 more figures

Theorems & Definitions (7)

  • Example 3.1: 1D ODE with one boundary layer
  • Example 3.2: 1D ODE with two boundary layers
  • Example 3.3: 1D viscous Burgers' equation
  • Example 3.4: 2D steady-state convection-diffusion problem
  • Example 3.5: 2D Helmholtz problem
  • Example 3.6: 1D ODE with one boundary layer, comparison with multi-level neural networks
  • Example 3.7: 1D steady-state Burgers' equation