We propose a particle method for numerically solving the Landau equation,
inspired by the score-based transport modeling (SBTM) method for the
Fokker-Planck equation. This method can preserve some important physical
properties of the Landau equation, such as the conservation of mass, momentum,
and energy, and decay of estimated entropy. We prove that matching the gradient
of the logarithm of the approximate solution is enough to recover the true
solution to the Landau equation with Maxwellian molecules. Several numerical
experiments in low and moderately high dimensions are performed, with
particular emphasis on comparing the proposed method with the traditional
particle or blob method.

我们提出了一种用于数值求解 Landau 方程的粒子方法，灵感来自于 Fokker-Planck 方程的基于得分的输运建模方法。该方法可以保持 Landau 方程的一些重要物理特性，例如质量、动量和能量的守恒性，以及估计熵的衰减。我们证明，匹配近似解的对数梯度足以恢复具有 Maxwellian 分子的 Landau 方程的真实解。在低维和中等高维上进行了几个数值实验，特别强调了所提出的方法与传统粒子或斑点方法的比较。