When surface evolution meets Fokker-Planck equation: a novel tangential velocity model for uniform parametrization

A common issue in simulating geometric evolution of surfaces is unexpected clustering of points that may cause numerical instability. We propose a novel artificial tangential velocity method for this matter. The artificial tangential velocity is generated from a surface density field governed by a Fokker-Planck equation to guide the point distribution. A target distribution matching algorithm is developed leveraging the surface Kullback-Leibler divergence of density functions. The numerical method is formulated within a fully meshless framework using the moving least squares approximation, thereby eliminating the need for mesh generation and allowing flexible treatment of unstructured point cloud data. Extensive numerical experiments are conducted to demonstrate the robustness, accuracy, and effectiveness of the proposed approach across a variety of surface evolution problems, including the mean curvature flow.