基于神经评分函数的Vlasov-Maxwell-Landau系统粒子方法

摘要

等离子体建模是核聚变反应堆设计的核心，然而基于第一性原理模拟碰撞等离子体动力学仍面临巨大的计算挑战：弗拉索夫-麦克斯韦-朗道（VML）系统描述了六维相空间在自洽电磁场作用下的输运过程，同时包含非线性、非局域的朗道碰撞算子。近期针对完整VML系统的确定性粒子方法通过团簇法估计速度评分函数，这种基于核函数的近似方法具有O(n²)计算复杂度。本研究采用基于评分的输运建模（SBTM）替代团簇评分估计器，该方法通过隐式评分匹配以O(n)复杂度实时训练神经网络。我们证明了近似碰撞算子能保持动量和动能守恒，并耗散估计熵值。同时刻画了VML系统及其静电简化形式的唯一全局稳态，为数值验证提供基准真值。在朗道阻尼、双流不稳定性、韦贝尔不稳定性三个经典基准测试中，SBTM相比团簇法精度更高，能正确实现团簇法失效的长时间弛豫至麦克斯韦平衡态的过程，且运行速度提升50%，峰值内存降低4倍。

English

Plasma modeling is central to the design of nuclear fusion reactors, yet simulating collisional plasma kinetics from first principles remains a formidable computational challenge: the Vlasov-Maxwell-Landau (VML) system describes six-dimensional phase-space transport under self-consistent electromagnetic fields together with the nonlinear, nonlocal Landau collision operator. A recent deterministic particle method for the full VML system estimates the velocity score function via the blob method, a kernel-based approximation with O(n^2) cost. In this work, we replace the blob score estimator with score-based transport modeling (SBTM), in which a neural network is trained on-the-fly via implicit score matching at O(n) cost. We prove that the approximated collision operator preserves momentum and kinetic energy, and dissipates an estimated entropy. We also characterize the unique global steady state of the VML system and its electrostatic reduction, providing the ground truth for numerical validation. On three canonical benchmarks -- Landau damping, two-stream instability, and Weibel instability -- SBTM is more accurate than the blob method, achieves correct long-time relaxation to Maxwellian equilibrium where the blob method fails, and delivers 50% faster runtime with 4times lower peak memory.

基于神经评分函数的Vlasov-Maxwell-Landau系统粒子方法

A Neural Score-Based Particle Method for the Vlasov-Maxwell-Landau System

摘要

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