Macroscopic auxiliary asymptotic preserving neural networks for the
  linear radiative transfer equations
Hongyan Li Song Jiang Wenjun Sun Liwei Xu Guanyu Zhou
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.