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Temporal Lifting as Latent-Space Regularization for Continuous-Time Flow Models in AI Systems

Jeffrey Camlin

TL;DR

This work introduces temporal lifting, an adaptive latent-space time-warping approach that maps $t$ to a smooth, strictly increasing $\tilde{t}$ via $\tilde{t}=\varphi(t)$ and defines lifted fields $U(x,\tau)=u(x,\varphi(\tau))$, to regularize derivative discontinuities in continuous-time flow models such as the incompressible Navier--Stokes equations on $\mathbb{T}^3$. The main result, the Temporal Lift Equivalence Theorem, shows that the lifted system $\varphi'(\tau)\partial_\tau U + (U\cdot\nabla)U + \nabla P - \nu\Delta U=0$ preserves Leray--Hopf energy structure and key regularity criteria, with invariance of Prodi--Serrin and Beale--Kato--Majda blowup conditions; singularities may be shifted if $\varphi'$ is allowed to vanish or diverge. Numerical validation on a $256^3$ grid with Taylor--Green initial data confirms energy conservation and the Beale--Kato--Majda criterion across physical and lifted coordinates, supporting coordinate-independence of blowup diagnostics. The framework connects analytic regularity theory with representation-learning techniques (e.g., latent-space dynamics, neural ODEs, PINNs) to stabilize continuous-time models used in AI systems, potentially improving global regularity and robustness in physics-informed learning contexts.

Abstract

We present a latent-space formulation of adaptive temporal reparametrization for continuous-time dynamical systems. The method, called *temporal lifting*, 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 or time-warping 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.

Temporal Lifting as Latent-Space Regularization for Continuous-Time Flow Models in AI Systems

TL;DR

This work introduces temporal lifting, an adaptive latent-space time-warping approach that maps to a smooth, strictly increasing via and defines lifted fields , to regularize derivative discontinuities in continuous-time flow models such as the incompressible Navier--Stokes equations on . The main result, the Temporal Lift Equivalence Theorem, shows that the lifted system preserves Leray--Hopf energy structure and key regularity criteria, with invariance of Prodi--Serrin and Beale--Kato--Majda blowup conditions; singularities may be shifted if is allowed to vanish or diverge. Numerical validation on a grid with Taylor--Green initial data confirms energy conservation and the Beale--Kato--Majda criterion across physical and lifted coordinates, supporting coordinate-independence of blowup diagnostics. The framework connects analytic regularity theory with representation-learning techniques (e.g., latent-space dynamics, neural ODEs, PINNs) to stabilize continuous-time models used in AI systems, potentially improving global regularity and robustness in physics-informed learning contexts.

Abstract

We present a latent-space formulation of adaptive temporal reparametrization for continuous-time dynamical systems. The method, called *temporal lifting*, introduces a smooth monotone mapping 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 become globally smooth. From the standpoint of machine-learning dynamics, temporal lifting acts as a continuous-time normalization or time-warping 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.

Paper Structure

This paper contains 8 sections, 1 theorem, 11 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Temporal lifting and motivation
  3. Preliminaries
  4. Function Spaces and Navier--Stokes Equations
  5. Temporal Lifting
  6. Main Theorem
  7. Numerical Validation
  8. Example Algorithm

Key Result

Theorem 3.1

Let $u(x,t)$ be a Leray--Hopf (resp. classical) solution of the incompressible Navier--Stokes equations on the three--torus $\mathbb{T}^3 = \mathbb{R}^3/\mathbb{Z}^3$, given in eq:NSE--eq:div of Section subsec2.1, with initial data $u_0(x)\in H^s_{\mathrm{div}}(\mathbb{T}^3)$ defined in eq:divspace. as introduced in eq:liftdef of Section subsec2.2. Then $U$ is a Leray--Hopf (resp. classical) solut

Figures (1)

  • Figure 1: Temporal lifting as a geometric analogy. The blue trajectory $\gamma(t)$ on $S^1$ develops a discontinuity at $T^*$ (red arc), while the lifted helix $\tilde{\gamma}(\theta)$ on $\mathbb{R}$ (green) is globally smooth, mapping the pre- and post-lift points at $\theta=0$, $\theta=2\pi$.

Theorems & Definitions (2)

  • Theorem 3.1: Temporal Lift Equivalence Theorem
  • proof