用变分粒子法模拟等离子体碰撞方程，兼具稳定性和高效率。
JKO for Landau: a variational particle method for homogeneous Landau equation
- 基于JKO框架设计隐式粒子方法，利用得分函数重构流场映射。
- 实现精确熵耗散与无条件稳定，支持长时间大尺度模拟。
- 专为随机优化设计，显著降低计算复杂度，适合大规模仿真。
受Landau方程梯度流视角及[arXiv:2007.08591]中对应的动态形式启发，本文在JKO框架下提出一种新型隐式粒子方法求解均匀Landau方程。首先将Landau度量转化为计算友好的形式，并通过流映射将其转换至拉格朗日视角。关键观察发现：尽管流映射遵循复杂的积分方程演化，其未知部分仅为对应密度的得分函数外加一个属于碰撞核零空间的附加项。这一洞察指导了流映射神经网络的设计与训练。此外，目标函数呈双重求和形式，极适于随机方法。因此，我们设计了一种定制化随机梯度下降算法，保持粒子间相互作用的同时大幅降低计算复杂度。相较于其他确定性粒子方法，该方法具有精确熵耗散与无条件稳定性，适用于长时间、大尺度等离子体模拟。
原文摘要 · Abstract (English)
Inspired by the gradient flow viewpoint of the Landau equation and the corresponding dynamic formulation of the Landau metric in [arXiv:2007.08591], we develop a novel implicit particle method for the Landau equation in the framework of the JKO scheme. We first reformulate the Landau metric in a computationally friendly form, and then translate it into the Lagrangian viewpoint using the flow map. A key observation is that, while the flow map evolves according to a rather complicated integral equation, the unknown component is simply a score function of the corresponding density plus an additional term in the null space of the collision kernel. This insight guides us in designing and training the neural network for the flow map. Additionally, the objective function is in a double summation form, making it highly suitable for stochastic methods. Consequently, we design a tailored version of stochastic gradient descent that maintains particle interactions and significantly reduces the computational complexity. Compared to other deterministic particle methods, the proposed method enjoys exact entropy dissipation and unconditional stability, therefore making it suitable for large-scale plasma simulations over extended time periods.
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