通过时间提升让连续时间流模型更稳定，尤其适合处理湍流等复杂系统。
Temporal Lifting as Latent-Space Regularization for Continuous-Time Flow Models in AI Systems
- 用平滑单调的时间映射修复流体方程的奇异行为
- 在提升空间中，拓扑环面上的流动轨迹全局光滑
- 适用于物理信息神经网络和复杂动力系统的建模
我们提出了一种连续时间动力系统中的自适应时间提升的隐空间形式。该方法引入一个平滑单调的映射 $t \mapsto τ(t)$，在保持守恒律的同时，正则化底层流的近奇异行为。在提升坐标下，如在环面 $ℤ^3$ 上的不可压缩纳维-斯托克斯方程的轨迹变得全局光滑。从机器学习动力学的角度看，时间提升相当于一种连续时间归一化算子，可稳定物理信息神经网络及其他用于人工智能系统的隐流架构。该框架将解析正则性理论与刚性或湍流过程的表示学习方法相连接。
原文摘要 · Abstract (English)
We present a latent-space formulation of adaptive temporal lifting for continuous-time dynamical systems. The method introduces a smooth monotone mapping $t \mapsto τ(t)$ that regularizes near-singular behavior of the underlying flow while preserving its conservation laws. In the lifted coordinate, trajectories such as those of the incompressible Navier-Stokes equations on the torus $\mathbb{T}^3$ become globally smooth. From the standpoint of machine-learning dynamics, temporal lifting acts as a continuous-time normalization operator that can stabilize physics-informed neural networks and other latent-flow architectures used in AI systems. The framework links analytic regularity theory with representation-learning methods for stiff or turbulent processes.
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