基於神經網絡評分的粒子方法求解弗拉索夫-麥克斯韋-朗道方程組

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

March 26, 2026

作者: Vasily Ilin, Jingwei Hu

cs.AI

摘要

電漿建模是核融合反應爐設計的核心，然而從第一性原理模擬碰撞性電漿動力學仍是巨大的計算挑戰：Vlasov-Maxwell-Landau（VML）系統描述了六維相空間中自洽電磁場作用下的傳輸過程，以及非線性、非局域的朗道碰撞算子。近期針對完整VML系統的確定性粒子方法通過斑點法估算速度評分函數，這是一種基於核函數的近似方法，計算成本為O(n²)。本研究採用基於評分的傳輸建模（SBTM）替代斑點評分估算器，該方法通過隱式評分匹配以O(n)成本即時訓練神經網絡。我們證明了近似碰撞算子能保持動量與動能守恆，並耗散估算熵。同時刻畫了VML系統及其靜電簡化形式唯一的全局穩態，為數值驗證提供基準真相。在朗道阻尼、雙流不穩定性和韋伯不穩定性三個經典基準測試中，SBTM不僅精度優於斑點法，更在斑點法失效的場景下實現了正確的長時間馬克士威平衡弛豫，運行時間縮短50%，峰值記憶體用量降低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.