Vortex Stretching in the Navier-Stokes Equations and Information Dissipation in Diffusion Models: A Reformulation from a Partial Differential Equation Viewpoint
Tsuyoshi Yoneda
TL;DR
The paper addresses how coherent vortex structures can be traced backward in time in axisymmetric Navier–Stokes flows and investigates how information about the initial vorticity dissipates under strain. It builds a PDE-based reformulation of diffusion-model dynamics, deriving a time-reversed FP with a drift that encapsulates the time-reversed score of the density and absorbs the backward Laplacian, implemented via a discrete Lagrangian flow and a neural network to learn the score. The authors provide explicit derivations for the axisymmetric vorticity equation, extend the framework to FP with reaction terms, and establish a practical learning pipeline that yields backward trajectories from forward data. Their numerical experiments show anisotropic information dissipation: rapid loss in the compressive radial direction and better preservation along the stretching axial direction, offering an information-theoretic perspective on vortex dynamics and a path toward generalizing to broader turbulent flows.
Abstract
We present a new inverse-time formulation of vortex stretching in the Navier-Stokes equations, based on a PDE framework inspired by score-based diffusion models. By absorbing the ill-posed backward Laplacian arising from time reversal into a drift term expressed through a score function, the inverse-time dynamics are formulated in a Lagrangian manner. Using a discrete Lagrangian flow of an axisymmetric vortex-stretching field, the score function is learned with a neural network and employed to construct backward-time particle trajectories. Numerical results demonstrate that information about initial positions is rapidly lost in the compressive direction, whereas it is relatively well preserved in the stretching direction.