# A score-based particle method for homogeneous Landau equation

arXiv: https://arxiv.org/abs/2405.05187 · HTML: https://mlsanity.com/paper/2405.05187/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 51 references that are in the map.
- First posted on arXiv 8 May 2024.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [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)
- [Self-Consistent Velocity Matching of Probability Flows](https://mlsanity.com/paper/2301.13737/) (2023)
- [Score-based Transport Modeling for Mean-Field Fokker-Planck Equations](https://mlsanity.com/paper/2305.03729/) (2023)
- [Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization](https://mlsanity.com/paper/2205.13098/) (2022)
- [Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs](https://mlsanity.com/paper/2303.11205/) (2023)

## Built on by (most cited first, 6 of 6)

- [Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity](https://mlsanity.com/paper/2405.15986/) (2024)
- [What's the score? Automated Denoising Score Matching for Nonlinear Diffusions](https://mlsanity.com/paper/2407.07998/) (2024)
- [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)
- [JKO for Landau: a variational particle method for homogeneous Landau equation](https://mlsanity.com/paper/2409.12296/) (2024)
- [A Neural Score-Based Particle Method for the Vlasov-Maxwell-Landau System](https://mlsanity.com/paper/2603.25832/) (2026)

## Abstract (arXiv)

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.