AI 中文总结
本文提出一种粒子方法，通过从提升线性算子格林函数摊销采样求解玻尔兹曼方程，复杂度为O(N)，可严格守恒动量能量，能直接从散射数据学习格林函数。
AI 中文摘要
玻尔兹曼方程的碰撞算子是一种非线性非局部算子，当在扩展的双粒子空间中提升时，可视为碰撞线性算子的投影。本文提出一种粒子方法，直接从该算子的格林函数（生成的时间连续马尔可夫链的转移概率）中采样碰撞后的相对速度；随后提出归一化流摊销采样以降低采样复杂度。该方法每次计算复杂度为O(N)（N为粒子数），可严格保持动量和能量守恒，无需核函数有界；更重要的是，它允许直接从散射数据中学习格林函数，无需在指定族中选择核函数。
英文摘要
The collision operator for the Boltzmann equation is a nonlinear nonlocal operator. When lifted in the extended 2-particle space, it is viewed as the projection of a collisional linear operator. In this paper, we propose a particle method that samples the post-collision relative velocity directly from the Green's function of this operator (the transition probability of the generated time-continuous Markov chain). The normalizing flow amortized sampling is then proposed to reduce the sampling complexity. The resulted method takes $O(N)$ each time where $N$ is the particle number, and conserves momentum and energy exactly. This method does not require the boundedness of the kernel and, more importantly, it allows learning the Green's function directly from the scattering data without selecting the kernel in a specified family.
Comments30 pages, 12 figures
