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A score-based particle method for homogeneous Landau equation

Yan Huang, Li Wang

TL;DR

The paper tackles the computational challenge of the space-homogeneous Landau equation, where the nonlinear velocity field depends on the density through the score of the distribution. It introduces a score-based particle method that reformulates the equation as a continuity equation and uses a transport-map (flow map) view to evolve particles while learning the score function $\nabla \log \tilde{f}_t$ with score-matching, thereby avoiding costly kernel density estimation. A key theoretical result shows that the KL divergence between the learned density and the true solution can be controlled by the score-matching loss, and an exact density update along particle trajectories is derived via the determinant of the transport map. Numerical experiments for Maxwell and Coulomb interactions demonstrate conservation properties, entropy dissipation, and substantial computational efficiency gains, with clear pathways to extending the approach to inhomogeneous settings and further speedups with batching or tree-based methods.

Abstract

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.

A score-based particle method for homogeneous Landau equation

TL;DR

The paper tackles the computational challenge of the space-homogeneous Landau equation, where the nonlinear velocity field depends on the density through the score of the distribution. It introduces a score-based particle method that reformulates the equation as a continuity equation and uses a transport-map (flow map) view to evolve particles while learning the score function with score-matching, thereby avoiding costly kernel density estimation. A key theoretical result shows that the KL divergence between the learned density and the true solution can be controlled by the score-matching loss, and an exact density update along particle trajectories is derived via the determinant of the transport map. Numerical experiments for Maxwell and Coulomb interactions demonstrate conservation properties, entropy dissipation, and substantial computational efficiency gains, with clear pathways to extending the approach to inhomogeneous settings and further speedups with batching or tree-based methods.

Abstract

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.
Paper Structure (16 sections, 8 theorems, 101 equations, 11 figures, 2 algorithms)

This paper contains 16 sections, 8 theorems, 101 equations, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A score-based particle method
  3. Lagrangian formulation via transport map
  4. Learning the score
  5. Algorithm
  6. Theoretical analysis
  7. Exact density computation
  8. Numerical results
  9. Example 1: 2D BKW solution for Maxwell molecules
  10. Example 2: 3D BKW solution for Maxwell molecules
  11. Example 3: 2D anisotropic solution with Coulomb potential
  12. Example 4: 3D Rosenbluth problem with Coulomb potential
  13. Density computation via Algorithm \ref{['algorithm_densitiy']}
  14. Conclusion
  15. Proof of Proposition \ref{['prop:pd']}
  16. ...and 1 more sections

Key Result

Proposition 2.1

The particle formulation particle0 conserves mass, momentum, and energy. Specifically, the following quantities remain constant over time:

Figures (11)

  • Figure 1: Score visualization. The first two rows are at $t=1$, the last two rows are at $t=5$.
  • Figure 2: The location of particles at $t=0, 1$, and $5$
  • Figure 3: Quantitative comparisons between the numerical solution and analytical solution.
  • Figure 4: Slices of the reconstructed and analytical solutions at $t=1$, $2.5$, and $5$.
  • Figure 5: Quantitative comparisons between the numerical solution and analytical solution.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 1
  • Proposition 3.1
  • proof
  • Theorem 3.2: Time evolution of the KL divergence on $\mathbb{T}^d$
  • Lemma 3.3
  • proof
  • ...and 11 more