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Transport based particle methods for the Fokker-Planck-Landau equation

Vasily Ilin, Jingwei Hu, Zhenfu Wang

TL;DR

The paper addresses the numerical solution of the spatially homogeneous Landau equation by extending score-based transport modeling (SBTM) to the Landau operator. It introduces a particle method that substitutes the score $\nabla\log u_t$ with a learned $s_t$ and propagates particles via $\frac{dX_i}{dt}=v_t[s_t](X_i)$, proving a KL-divergence bound that relates score-matching error to convergence when dealing with Maxwellian molecules. The approach preserves mass, momentum, and energy while maintaining entropy dissipation, and numerical experiments demonstrate advantages over blob methods, particularly for anisotropic and high-dimensional problems, with favorable runtimes and GPU scalability. Potential extensions include handling Coulomb interactions ($\gamma=-3$), adaptive training strategies, and applying the framework as a kernel for the full Vlasov–Landau equation, broadening its practical impact for kinetic plasma simulations.

Abstract

We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.

Transport based particle methods for the Fokker-Planck-Landau equation

TL;DR

The paper addresses the numerical solution of the spatially homogeneous Landau equation by extending score-based transport modeling (SBTM) to the Landau operator. It introduces a particle method that substitutes the score with a learned and propagates particles via , proving a KL-divergence bound that relates score-matching error to convergence when dealing with Maxwellian molecules. The approach preserves mass, momentum, and energy while maintaining entropy dissipation, and numerical experiments demonstrate advantages over blob methods, particularly for anisotropic and high-dimensional problems, with favorable runtimes and GPU scalability. Potential extensions include handling Coulomb interactions (), adaptive training strategies, and applying the framework as a kernel for the full Vlasov–Landau equation, broadening its practical impact for kinetic plasma simulations.

Abstract

We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.
Paper Structure (14 sections, 11 theorems, 101 equations, 6 figures)

This paper contains 14 sections, 11 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Properties of the Landau equation
  3. Particle methods for the Landau equation
  4. Score Based Transport Modeling (SBTM)
  5. Kullback–Leibler divergence bound
  6. KL bound for the Fokker-Planck equation
  7. KL bound for the Landau equation
  8. Numerical experiments
  9. Hyperparameter choice
  10. Landau equation with Maxwellian kernel
  11. Landau equation with Coulomb kernel
  12. Runtime
  13. Conclusion
  14. Appendix: The continuity equation and its particle solution

Key Result

Theorem 2.1

In case of the Maxwellian molecules ($\gamma=0$ in eqn: Landau collision kernel), if the initial data $u_0$ has finite energy the Landau equation eqn: Landau equation1 has a unique solution $u_t$ defined for all $t\geq 0$. Moreover, for all $t>0$, $u_t$ is bounded and belongs to $C^\infty(\mathbb{R}^d)$. If $u_0$ is subgaussian, then $u_t$ is subgaussian for all $t$: In case of the Coulomb inter

Figures (6)

  • Figure 1: Effect of underfitting, optimal $\eta=4\cdot10^{-4}$ (left) vs low $\eta=10^{-4}$ (right) and non-adaptive training strategy (top) vs adaptive training strategy (bottom).
  • Figure 2: (An isotropic) BKW solution to the Landau equation with Maxwellian kernel in dimension $d=3$. Both the blob method and SBTM do a decent job of approximating the solution.
  • Figure 3: (An anisotropic) solution to the Landau equation with Maxwellian kernel in dimension $d=3$. SBTM can match the covariance well even with $n=100$ particles, while the blob methods needs a lot more particles to achieve the same accuracy.
  • Figure 4: (An anisotropic) solution to the Landau equation with Maxwellian kernel in dimension $d=10$. Even $n=25600$ particles are not enough for the blob method to match the true covariance trajectory, whereas SBTM gives a very good prediction.
  • Figure 5: (An anisotropic) solution to the Landau equation with Coulomb kernel in dimension $d=3$. SBTM achieves better accuracy with $n=100$ particles in terms of matching covariance and entropy decay rate of the simulations with $n=12800$ particles.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 2.1: Existence, uniqueness, and regularity
  • Proposition 2.2: Fokker-Planck form
  • proof
  • Lemma 2.3: Boundedness of $A\ast u$
  • proof
  • Theorem 2.4
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 17 more