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

arXiv: https://arxiv.org/abs/2603.25832 · HTML: https://mlsanity.com/paper/2603.25832/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 26 Mar 2026.

## Builds on (most cited first, 9 of 9)

- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Self-Consistency of the Fokker-Planck Equation](https://mlsanity.com/paper/2206.00860/) (2022)
- [Score-based Transport Modeling for Mean-Field Fokker-Planck Equations](https://mlsanity.com/paper/2305.03729/) (2023)
- [Semi-Autonomous Formalization of the Vlasov-Maxwell-Landau Equilibrium](https://mlsanity.com/paper/2603.15929/) (2026)
- [A score-based particle method for homogeneous Landau equation](https://mlsanity.com/paper/2405.05187/) (2024)
- [DiScoFormer: Plug-In Density and Score Estimation with Transformers](https://mlsanity.com/paper/2511.05924/) (2025)
- [Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs](https://mlsanity.com/paper/2303.11205/) (2023)
- [Transport based particle methods for the Fokker-Planck-Landau equation](https://mlsanity.com/paper/2405.10392/) (2024)
- [Score-based deterministic density sampling](https://mlsanity.com/paper/2504.18130/) (2025)

## Abstract (arXiv)

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(n2) 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 4× lower peak memory.