Vortex Stretching in the Navier-Stokes Equations and Information Dissipation in Diffusion Models: A Reformulation from a Partial Differential Equation Viewpoint

TL;DR

This paper introduces a PDE-based inverse-time formulation of vortex stretching in Navier-Stokes equations, inspired by diffusion models, and uses neural networks to analyze information dissipation in fluid dynamics.

Contribution

It reformulates vortex stretching as an inverse-time PDE problem and employs neural networks to learn score functions for backward trajectory reconstruction.

Findings

Information rapidly lost in compressive directions

Information preserved in stretching directions

Neural network effectively models inverse dynamics

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.