Speaker
Description
We propose a physics-informed neural particle method (PINN-PM) for the spatially homogeneous Landau equation. The method adopts a Lagrangian interacting-particle formulation and jointly parameterizes the time-dependent score and the characteristic flow map with neural networks. Instead of advancing particles through explicit time stepping, the Landau dynamics is enforced via a continuous-time residual defined along particle trajectories, yielding a mesh-free solver that can be queried at arbitrary times. We establish a deterministic, global-in-time stability analysis in an $L^2_v$ framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: the score approximation error, the particle approximation error, and the physics residual of the neural flow. This trajectory estimate is then lifted to Wasserstein stability and density reconstruction error via kernel density estimation, resulting in a complete error propagation chain from particle dynamics to macroscopic quantities. At the oracle level, the score error is characterized through the implicit score matching functional via Hyv\"arinen's identity. In practice, the empirical ISM objective provides a computable surrogate for monitoring score accuracy during training. Numerical experiments on analytical benchmarks, including the two- and three-dimensional BKW solutions, as well as reference-free configurations, demonstrate stable transport, preservation of macroscopic invariants, and competitive or improved accuracy compared with time-stepping particle methods while using significantly fewer particles.