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

• URL: http://arxiv.org/abs/2403.01820v1
• Date: Mon, 4 Mar 2024 08:10:42 GMT
• Title: Macroscopic auxiliary asymptotic preserving neural networks for the linear radiative transfer equations
• Authors: Hongyan Li, Song Jiang, Wenjun Sun, Liwei Xu, Guanyu Zhou
• Abstract summary: We develop a Macroscopic Auxiliary Asymptotic-Preserving Neural Network (MA-APNN) method to solve the time-dependent linear radiative transfer equations. We present several numerical examples to demonstrate the effectiveness of MA-APNNs.
• Score: 3.585855304503951
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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