A score-based particle method for homogeneous Landau equation

The Landau equation stands as one of the fundamental kinetic equations, modeling the evolution of charged particles undergoing Coulomb interaction [27]. It is particularly useful for plasmas where collision effects become non-negligible. Computing the Landau equation presents numerous challenges inherent in kinetic equations, including high dimensionality, multiple scales, and strong nonlinearity and non-locality. On the other hand, deep learning has progressively transformed the numerical computation of partial differential equations by leveraging neural networks' ability to approximate complex functions and the powerful optimization toolbox. However, straightforward application of deep learning to compute PDEs often encounters training difficulties and leads to a loss of physical fidelity. In this paper, we propose a score-based particle method that elegantly combines learning with structure-preserving particle methods. This method inherits the favorable conservative properties of deterministic particle methods while relying only on light training to dynamically obtain the score function over time. The learning component replaces the expensive density estimation in previous particle methods, drastically accelerating computation.