本文提出了渐近保持随机特征方法（Asymptotic-Preserving Random Feature Method，APRFM），用于高效求解多尺度辐射输运方程。APRFM通过采用微宏观分解策略，有效解决了辐射输运方程固有的刚性和多尺度特性带来的挑战。该方法将分布函数分解为平衡分量和非平衡分量，在最小二乘最小化框架内通过随机特征方法（random feature method，RFM）对两部分进行近似。所提方法在不同尺度下均表现出显著的鲁棒性，且与原始RFM相比，能以更少的自由度和配点达到高精度。此外，与基于深度神经网络的方法相比，本文方法在参数效率和计算速度方面具有显著优势。这些优势已通过针对一维和二维问题开展的大量数值实验得到验证。
英文摘要：
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.
A concurrent global-local numerical method for multiscale parabolic equations
National University of Singapore（新加坡国立大学）；SKLMS, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
A level-set approach for a multi-scale cancer invasion model
Institute for Applied Mathematics, Heidelberg University（海德堡大学应用数学研究所）；Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University（海德堡大学跨学科科学计算中心）；Division of Mathematics, University of Dundee（邓迪大学数学系）
