  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.
