Macroscopic auxiliary asymptotic preserving neural networks for the linear radiative transfer equations
This work addresses computational difficulties in radiative transfer for physics and engineering applications, but it appears incremental as it builds on existing Physics-Informed Neural Networks with a new loss function.
The authors tackled solving time-dependent linear radiative transfer equations with multi-scale and high-dimensional challenges by developing a Macroscopic Auxiliary Asymptotic-Preserving Neural Network (MA-APNN) method, which demonstrated effectiveness in numerical examples.
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.