Vortex Stretching in the Navier-Stokes Equations and Information Dissipation in Diffusion Models: A Reformulation from a Partial Differential Equation Viewpoint
This work addresses fluid dynamics modeling for researchers, but it is incremental as it applies an existing method to a new domain.
The paper tackled the problem of vortex stretching in the Navier-Stokes equations by reformulating it using a PDE framework inspired by diffusion models, resulting in numerical findings that information about initial positions is rapidly lost in the compressive direction but preserved in the stretching direction.
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.