Weighted Laplacian Flow: A Deterministic Particle Flow with Provable Convergence

Abstract

Sampling from a target probability density is a fundamental task in statistics, machine learning, and scientific computing. We introduce weighted Laplacian flow, a deterministic particle-flow method that transports samples from a tractable initial density to a target density known up to normalization. The method evolves the logarithmic density ratio between the target and the transported distribution and constructs the particle velocity by solving a weighted Poisson equation associated with the target density. This design avoids the need to choose a kernel and enables direct control of particle weights along the flow. We establish the global well-posedness of the proposed PDE system and prove that the transported density converges to the target density in both $L^{\infty }$ distance and Kullback-Leibler divergence. Under a sublinear forcing condition, the method achieves exact convergence in finite time. Numerical experiments on multimodal, heavy-tailed, and ten-dimensional targets demonstrate that weighted Laplacian flow can perform long-range mass transport, overcome energy barriers.