A Long-Short Flow-Map Perspective for Drifting Models

TL;DR

This paper introduces a novel long-short flow-map decomposition for drifting models, providing a new perspective on transport processes, and proposes a likelihood learning method validated through theoretical and empirical analysis.

Contribution

It offers a new semigroup-consistent flow-map factorization for drifting models and a likelihood learning formulation aligned with density evolution.

Findings

Decomposition into long-horizon and short-time flow maps captures transport dynamics.

Likelihood learning aligns with density evolution under the flow-map framework.

Theoretical analysis supports the proposed decomposition and learning approach.

Abstract

This paper provides a reinterpretation of the Drifting Model~\cite{deng2026generative} through a semigroup-consistent long-short flow-map factorization. We show that a global transport process can be decomposed into a long-horizon flow map followed by a short-time terminal flow map admitting a closed-form optimal velocity representation, and that taking the terminal interval length to zero recovers exactly the drifting field together with a conservative impulse term required for flow-map consistency. Based on this perspective, we propose a new likelihood learning formulation that aligns the long-short flow-map decomposition with density evolution under transport. We validate the framework through both theoretical analysis and empirical evaluations on benchmark tests, and further provide a theoretical interpretation of the feature-space optimization while highlighting several open problems for future study.