We propose a novel score-based particle method for solving the Landau equation in plasmas, that seamlessly integrates learning with structure-preserving particle methods [arXiv:1910.03080]. Building upon the Lagrangian viewpoint of the Landau equation, a central challenge stems from the nonlinear dependence of the velocity field on the density. Our primary innovation lies in recognizing that this nonlinearity is in the form of the score function, which can be approximated dynamically via techniques from score-matching. The resulting method inherits the conservation properties of the deterministic particle method while sidestepping the necessity for kernel density estimation in [arXiv:1910.03080]. This streamlines computation and enhances scalability with dimensionality. Furthermore, we provide a theoretical estimate by demonstrating that the KL divergence between our approximation and the true solution can be effectively controlled by the score-matching loss. Additionally, by adopting the flow map viewpoint, we derive an update formula for exact density computation. Extensive examples have been provided to show the efficiency of the method, including a physically relevant case of Coulomb interaction.

翻译：我们提出了一种新颖的基于分数的粒子方法，用于求解等离子体中的Landau方程，该方法将学习过程与保结构粒子方法[arXiv:1910.03080]无缝集成。基于Landau方程的拉格朗日视角，一个核心挑战源于速度场对密度的非线性依赖。我们的主要创新在于认识到这种非线性以得分函数的形式存在，可通过得分匹配技术动态逼近。由此产生的方法继承了确定性粒子方法的守恒性质，同时避免了[arXiv:1910.03080]中核密度估计的必要性。这简化了计算并增强了维度可扩展性。此外，我们通过证明近似解与真实解之间的KL散度能够被得分匹配损失有效控制，给出了理论估计。进一步地，采用流图视角，我们推导了用于精确密度计算的更新公式。通过大量示例（包括物理相关的库仑相互作用情形）验证了该方法的有效性。