Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic-Preserving Neural Networks based on Even-odd Decomposition for Multiscale Gray Radiative Transfer Equations

Keke Wu, Xizhe Xie, Wengu Chen, Han Wang, Zheng Ma

TL;DR

This APNN method alleviates the rigorous conservation requirements while simultaneously incorporating an auxiliary deep neural network, distinguishing it from the APNN method based on micro-macro decomposition for GRTE.

Abstract

We present a novel Asymptotic-Preserving Neural Network (APNN) approach utilizing even-odd decomposition to tackle the nonlinear gray radiative transfer equations (GRTEs). Our AP loss demonstrates consistent stability concerning the small Knudsen number, ensuring the neural network solution uniformly converges to the diffusion limit solution. This APNN method alleviates the rigorous conservation requirements while simultaneously incorporating an auxiliary deep neural network, distinguishing it from the APNN method based on micro-macro decomposition for GRTE. Several numerical problems are examined to demonstrate the effectiveness of our proposed APNN technique.

Asymptotic-Preserving Neural Networks based on Even-odd Decomposition for Multiscale Gray Radiative Transfer Equations

TL;DR

This APNN method alleviates the rigorous conservation requirements while simultaneously incorporating an auxiliary deep neural network, distinguishing it from the APNN method based on micro-macro decomposition for GRTE.

Abstract

We present a novel Asymptotic-Preserving Neural Network (APNN) approach utilizing even-odd decomposition to tackle the nonlinear gray radiative transfer equations (GRTEs). Our AP loss demonstrates consistent stability concerning the small Knudsen number, ensuring the neural network solution uniformly converges to the diffusion limit solution. This APNN method alleviates the rigorous conservation requirements while simultaneously incorporating an auxiliary deep neural network, distinguishing it from the APNN method based on micro-macro decomposition for GRTE. Several numerical problems are examined to demonstrate the effectiveness of our proposed APNN technique.
Paper Structure (14 sections, 43 equations, 11 figures, 4 tables)

This paper contains 14 sections, 43 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. The gray radiative transfer equation and its diffusion limit
  4. Micro-macro decomposition method for GRTEs
  5. Even-odd decomposition method for GRTEs wuAPNNv2jin2000uniformly
  6. Solving GRTEs by DNNs
  7. Numerical results
  8. Experiment setting
  9. Problem 1: APNNs for solving the linear transport equation
  10. Problem 2: APNNs for solving the stationary nonlinear GRTE
  11. Problem 3: APNNs for solving the time-dependent nonlinear GRTE
  12. Problem 4: APNNs for solving another time-dependent GRTE
  13. Problem 5: APNNs for solving Marshak wave problem
  14. Conclusion

Figures (11)

  • Figure 1: The idea of APNN-MM and APNN-EO for solving the GRTEs.
  • Figure 2: Schematic plot of APNNs based on even-odd decomposition for solving the GRTEs.
  • Figure 3: Plot of density at time $t = 0.1$ ($\varepsilon = 10^{-3}$): Approximated by PINN, APNNs based on micro-macro and even-odd decomposition (marker) vs. reference solution (line).
  • Figure 4: Plot of density and temperature ($\varepsilon = 10^{-3}$): Approximated by APNN based on micro-macro decomposition (marker) vs. reference solutions (line). The relative $\ell^2$ error of $\rho$ and $T$ by APNNs based on micro-macro decomposition are $9.42 \times 10^{-4}, 5.09 \times 10^{-3}$.
  • Figure 5: Plot of density and temperature ($\varepsilon = 10^{-3}$): Approximated by APNN based on even-odd decomposition (marker) vs. reference solutions (line). The relative $\ell^2$ error of $\rho$ and $T$ by APNNs based on even-odd decomposition are $6.26 \times 10^{-4}, 9.12 \times 10^{-4}$.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Remark 1