A Physics-Informed, Global-in-Time Neural Particle Method for the Spatially Homogeneous Landau Equation

TL;DR

This paper introduces a novel physics-informed neural particle method for the Landau equation that avoids time discretization errors, provides stability analysis, and demonstrates improved accuracy with fewer particles in numerical benchmarks.

Contribution

The paper presents a continuous-time, mesh-free neural particle method for the Landau equation that jointly learns the score and flow map, removing time-stepping errors and enabling arbitrary-time queries.

Findings

Stable transport and preservation of invariants in numerical experiments.

Competitive or improved accuracy with fewer particles compared to existing methods.

Rigorous stability and error bounds established for the proposed method.

Abstract

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. This design removes time-discretization error and yields a mesh-free solver that can be queried at arbitrary times without sequential integration. We establish a rigorous stability analysis in an $L^2_v$ framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: (i) score approximation error, (ii) empirical particle approximation error, and (iii) the physics residual of the neural flow. This trajectory estimate propagates to density reconstruction, where we derive an $L^2_v$ error bound for kernel density estimators combining classical bias--variance terms with a trajectory-induced contribution. Using Hyvarinen's identity, we further relate the oracle score-matching gap to the $L^2_v$ score error and show that the empirical loss concentrates at the Monte Carlo rate, yielding computable a posteriori accuracy certificates. 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 score-based particle and blob methods while using significantly fewer particles.