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

arXiv 2403.01820 ↗First posted 4 Mar 2024math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

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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References · 18

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6 more
  1. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  2. 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
  3. 2021Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equationcited by 16
  4. 2019Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approachcited by 13
  5. 2023Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONetscited by 3
  6. 2022A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equationscited by 2

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