用扩散模型视角重写纳维-斯托克斯方程，揭示涡拉伸中信息耗散机制。
Vortex Stretching in the Navier-Stokes Equations and Information Dissipation in Diffusion Models: A Reformulation from a Partial Differential Equation Viewpoint
- 从反向时间出发，将涡拉伸建模为拉格朗日粒子轨迹的随机微分方程。
- 数值模拟显示初始位置信息在压缩方向快速丢失，拉伸方向相对保留。
- 适用于流体动力学与生成模型交叉研究者，尤其关注信息演化机制。
我们提出一种基于得分函数驱动的反向时间涡拉伸新形式，该方法受基于得分的扩散模型启发，将时间反转带来的病态逆拉普拉斯算子吸收进漂移项。通过轴对称涡拉伸场的离散拉格朗日流，利用神经网络学习得分函数，并用于构建反向时间粒子轨迹。数值结果表明，初始位置信息在压缩方向迅速丢失，而在拉伸方向则相对保持。该框架为理解流体中信息耗散提供了新的偏微分方程视角。
原文摘要 · Abstract (English)
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.
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