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Macroscopic auxiliary asymptotic preserving neural networks for the linear radiative transfer equations

Hongyan Li, Song Jiang, Wenjun Sun, Liwei Xu, Guanyu Zhou

TL;DR

A new adaptive exponentially weighted Asymptotic-Preserving Asymptotic-Preserving (AP) loss function, which consists of a macroscopic auxiliary equation taking into account the diffusion limit equation explicitly, is designed.

Abstract

We develop a Macroscopic Auxiliary Asymptotic-Preserving Neural Network (MA-APNN) method to solve the time-dependent linear radiative transfer equations (LRTEs), which have a multi-scale nature and high dimensionality. To achieve this, we utilize the Physics-Informed Neural Networks (PINNs) framework and design a new adaptive exponentially weighted Asymptotic-Preserving (AP) loss function, which incorporates the macroscopic auxiliary equation that is derived from the original transfer equation directly and explicitly contains the information of the diffusion limit equation. Thus, as the scale parameter tends to zero, the loss function gradually transitions from the transport state to the diffusion limit state. In addition, the initial data, boundary conditions, and conservation laws serve as the regularization terms for the loss. We present several numerical examples to demonstrate the effectiveness of MA-APNNs.

Macroscopic auxiliary asymptotic preserving neural networks for the linear radiative transfer equations

TL;DR

A new adaptive exponentially weighted Asymptotic-Preserving Asymptotic-Preserving (AP) loss function, which consists of a macroscopic auxiliary equation taking into account the diffusion limit equation explicitly, is designed.

Abstract

We develop a Macroscopic Auxiliary Asymptotic-Preserving Neural Network (MA-APNN) method to solve the time-dependent linear radiative transfer equations (LRTEs), which have a multi-scale nature and high dimensionality. To achieve this, we utilize the Physics-Informed Neural Networks (PINNs) framework and design a new adaptive exponentially weighted Asymptotic-Preserving (AP) loss function, which incorporates the macroscopic auxiliary equation that is derived from the original transfer equation directly and explicitly contains the information of the diffusion limit equation. Thus, as the scale parameter tends to zero, the loss function gradually transitions from the transport state to the diffusion limit state. In addition, the initial data, boundary conditions, and conservation laws serve as the regularization terms for the loss. We present several numerical examples to demonstrate the effectiveness of MA-APNNs.
Paper Structure (26 sections, 55 equations, 10 figures, 6 tables)

This paper contains 26 sections, 55 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Linear radiative transfer equations
  3. The boundary conditions
  4. The 1D and 2D linear models
  5. The uncertainty quantification (UQ) model
  6. The diffusion limit equation
  7. Macroscopic auxiliary asymptotic preserving neural networks
  8. The vanilla PINNs fail to resolve the multiscale LRTEs
  9. The macroscopic auxiliary asymptotic-preserving neural networks (MA-APNNs)
  10. The macroscopic auxiliary equation
  11. The exponentially weighted MA-APNNs loss function
  12. The MA-APNN empirical loss function
  13. Angular integration approximation
  14. Empirical loss functions, sample points, and labeled data
  15. Numerical Experiments
  16. ...and 11 more sections

Figures (10)

  • Figure 3.1: Schematic of MA-APNNs for solving the linear radiative transfer equation with initial and boundary data.
  • Figure 4.1: Kinetic regime with $\epsilon=1$. The density $\rho$ at times $t=0.15, 0.4, 1.0, 1.6, 4.0$. (Left) Ref v.s. PINNs. (Middle) Ref v.s. APNNs. (Right) Ref v.s. MA-APNNs.
  • Figure 4.2: Initial layer with $\epsilon=1$. The density $\rho$ at times $t=0.0, 0.1$. (Left) Ref v.s. MA-APNNs with boundary soft constraint. (Middle) Ref v.s. MA-APNNs with boundary hard constraint. (Right) Ref v.s. APNNs with boundary hard constraint.
  • Figure 4.3: Diffusion regime with $\epsilon=10^{-8}$. The density $\rho$ at times $t=0.01, 0.05, 0.15, 2.00$. (Left) Ref v.s. PINNs. (Middle) Ref v.s. APNNs. (Right) Ref v.s. MA-APNNs.
  • Figure 4.4: Diffusion regime with $\epsilon=10^{-4}$. The density $\rho$ at times $t=0.2, 0.4, 0.6, 0.8, 1.0$. (Left) Ref v.s. PINNs. (Middle) Ref v.s. APNNs. (Right) Ref v.s. MA-APNNs.
  • ...and 5 more figures