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

We develop a Macroscopic Auxiliary Asymptotic-Preserving Neural Network (MA-APNN) method to solve the time-dependent linear radiative transfer equations (LRTEs), which possess such features as multi-scale properties and high dimensionality. Utilizing the Physics-Informed Neural Networks (PINNs), we design a new adaptive exponentially weighted Asymptotic-Preserving (AP) loss function, which consists of a macroscopic auxiliary equation taking into account the diffusion limit equation explicitly. A novel property on the scheme is that the loss function gradually transfers from the transport state to the diffusion limit state as the scale parameter approaches zero. Furthermore, we combine the residual-based adaptive refinement (RAR) method with the MA-APNN method to obtain the MA-APNN-RAR method, which leads to an improvement in the accuracy and efficiency under some practical settings. A theoretical analysis about the approximation errors is given for the MA-APNN method, and numerical examples are presented to demonstrate the performance of MA-APNNs and MA-APNNs-RAR.