A Micro-Macro Decomposition-Based Asymptotic-Preserving Random Feature Method for Multiscale Radiative Transfer Equations

TL;DR

The paper tackles multiscale radiative transfer equations by developing APRFM, which combines a micro-macro decomposition with a random feature approximation to solve the RTE in a uniformly accurate, asymptotic-preserving manner. By writing $f(x,v)=\rho(x)+\varepsilon g(x,v)$ and solving a coupled macro-micro least-squares system where $\rho$ and $g$ are each represented with PoU-local random features, the method remains stable as $\varepsilon\to0$ and recovers the diffusion limit. Numerical experiments in 1D and 2D demonstrate that APRFM achieves high accuracy with far fewer degrees of freedom and collocation points than vanilla RFM and compares favorably to deep neural network approaches in both accuracy and speed, including challenging mixed-scale and highly anisotropic settings. The results indicate strong potential for applying APRFM to more complex kinetic problems and motivate future work on theoretical convergence, stability analysis, time dependence, and nonlinear extensions.

Abstract

This paper introduces the Asymptotic-Preserving Random Feature Method (APRFM) for the efficient resolution of multiscale radiative transfer equations. The APRFM effectively addresses the challenges posed by stiffness and multiscale characteristics inherent in radiative transfer equations through the application of a micro-macro decomposition strategy. This approach decomposes the distribution function into equilibrium and non-equilibrium components, allowing for the approximation of both parts through the random feature method (RFM) within a least squares minimization framework. The proposed method exhibits remarkable robustness across different scales and achieves high accuracy with fewer degrees of freedom and collocation points than the vanilla RFM. Additionally, compared to the deep neural network-based method, our approach offers significant advantages in terms of parameter efficiency and computational speed. These benefits have been substantiated through numerous numerical experiments conducted on both one- and two-dimensional problems.

Figures (13)

• Figure 1: Schematic diagram of the random feature method.
• Figure 2: Relative $\ell^{2}$ error of the RFM solution $f_{M}$ with respect to the degrees of freedom.
• Figure 3: Reference solution v.s. RFM solution.
• Figure 4: Schematic diagram of the asymptotic-preserving random feature method.
• Figure 5: Relative $\ell^{2}$ error of the APRFM solution with respect to the degrees of freedom.

Theorems & Definitions (1)

• Definition 1