BF-APNN: A Low-Memory Method for Accelerating the Solution of Radiative Transfer Equations

TL;DR

The paper addresses the computational burden of solving high dimensional, multiscale radiative transfer equations by introducing Basis Function Asymptotically Preserving Neural Network BF-APNN, which replaces costly angular integrations with basis function expansions for the microscopic component in a micro-macro decomposition. BF-APNN builds on RT-APNN by using basis expansions to represent g and by precomputing angular moments, enabling exact moments and preserving asymptotic diffusion limits while reducing training time and memory usage. Across linear, nonlinear, Marshak wave and 2D problems, BF-APNN achieves comparable or improved accuracy with substantial speedups over RT-APNN and other PINN-based methods, and demonstrates robustness to dimensionality and multiscale features. The approach holds promise for scalable GRTE solvers and suggests basis function strategies that can extend to other high dimensional PDEs.

Abstract

The Radiative Transfer Equations (RTEs) exhibit high dimensionality and multiscale characteristics, rendering conventional numerical methods computationally intensive. Existing deep learning methods perform well in low-dimensional or linear RTEs, but still face many challenges with high-dimensional or nonlinear RTEs. To overcome these challenges, we propose the Basis Function Asymptotically Preserving Neural Network (BF-APNN), a framework that inherits the advantages of Radiative Transfer Asymptotically Preserving Neural Network (RT-APNN) and accelerates the solution process. By employing basis function expansion on the microscopic component, derived from micro-macro decomposition, BF-APNN effectively mitigates the computational burden associated with evaluating high-dimensional integrals during training. Numerical experiments, which involve challenging RTE scenarios featuring, nonlinearity, discontinuities, and multiscale behavior, demonstrate that BF-APNN substantially reduces training time compared to RT-APNN while preserving high solution accuracy. Moreover, BF-APNN exhibits superior performance in addressing complex, high-dimensional RTE problems, underscoring its potential as a robust tool for radiative transfer computations.

Figures (11)

• Figure 3.1.1: Micro-macro network structure of BF-APNN. Here, $(t, \boldsymbol{x})$ represents the spatiotemporal variables, $(\theta, \varphi)$ represents the angular variables, $T_m$ is the material temperature (i.e., $T$ in the equation), and $T_r$ is the radiation temperature. $(\rho, g)$ represent the macro and micro components, respectively, in the micro-macro decomposition.
• Figure 3.3.1: A comparison plot of three 1D basis functions, with 12 functions plotted for each.
• Figure 4.3.1: The result of Ex 3. The material temperature at times $t=0.2\ \mathrm{ns}$, $0.4\ \mathrm{ns}$, $0.6\ \mathrm{ns}$, $0.8\ \mathrm{ns}$, $1.0\ \mathrm{ns}$.
• Figure 4.3.2: The result of Ex 3. Line plot of the microscopic component $g$ at different spatial positions at time $t=1.0\ \mathrm{ns}$.
• Figure 4.3.3: The result of Ex 3. The effect of the number of basis on the relative $L_2$ error.