A Long-Short Flow-Map Perspective for Drifting Models

Zhiqi Li; Bo Zhu

cs.LGFeb 24, 2026

A Long-Short Flow-Map Perspective for Drifting Models

Authors: Zhiqi Li, Bo Zhu

Organizations: School of Interactive Computing, Georgia Institute of Technology

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.

Figures & tables

Panel of Figure 1 — Figure 1: (a) Closed-form Flow Matching ( subsection 2.1 ) requires kernel-weighted aggregation over data points, which becomes prohibitively expensive when the kernel is diffuse (far from $t=1$ ). (b) Flow-map training ( subsection 2.2 ) enforces trajectory consistency but is difficult to supervise directly from data, so spurious consistent paths (e.g., the semi-transparent segment) can still satisfy the constraint. (c) Our Long-Short Flow Map method ( section 3 ) applies a closed form estimator on the short step $\psi_{1-\Delta t\to 1}$ as $\Delta t\to 0$ ( subsection 3.1 first order, subsection 3.2 second order), which provides dataset grounded supervision for learning the long step $\psi_{0\to 1-\Delta t}$ and thereby addresses the challenges of both closed form Flow Matching and flow map methods.

Figure 2: Overview of the Long-Short Flow Map framework. Leveraging flow-map trajectory consistency, we decompose the full map $\psi_{0\to 1}$ into a long map $\psi_{0\to 1-\Delta t}$ and a short map $\psi_{1-\Delta t\to 1}$ . The short map is approximated using forward Euler or the trapezoidal rule and then computed via the closed-form solution in flow matching, providing dataset-level supervision for learning the long map $\psi_{0\to 1-\Delta t}$ . Taking the limit $\Delta t\to 0$ yields a long-short flow-map derivation of the Drifting Model.

Figure 3: Unconditional latent generation results on CelebA-HQ. We train for 100K steps using feature-space optimization with a pretrained MAE as the feature extractor and achieve an FID of 14.71. For comparison, the corresponding MeanFlow model reaches an FID of 12.4 after 400K training steps. Full results are provided in Table 1 .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: 2D Examples. We show the ground-truth samples, generated samples, and learned likelihoods from the Eulerian and Lagrangian views on Spiral, Checkerboard, and Two Moons.

Panel of Figure 5 — Figure 5: Image generation. FID trajectories under different kernels and batch sizes, together with an ablation on using multiple feature spaces. The Laplacian kernel corresponds to the first-order kernel used in Drifting Model Deng et al. [2026] , whereas the Gaussian kernel is the second-order kernel derived in our framework. Overall, the Gaussian kernel performs comparably to the Laplacian kernel, and can be better in some regimes. In the ablation (batch size $B{=}64$ , lr $=5\times 10^{-5}$ ), we compare using the default feature set with thousands of channels, using only four features (the outputs at the four encoder resolutions), and operating directly in the original space; The latter two settings fail to produce high-quality results.

Figure 6: FID over training with a standard batch size. FID curves for the Long-Short Flow-Map method / Drifting Model over training, using batch size 64 (with a tuned learning rate). The results show that strong generative performance can be achieved without extremely large batch sizes.

Long-Short Flow Map / Drifting (ours)
Method	Batch size	Training steps	FID
Consistency Models Song et al. [2023]	64	400K	33.2
Shortcut Models Frans et al. [2025]	64	400K	20.5
MeanFlow Geng et al. [2025a]	64	400K	12.4
Laplacian kernel ( $B{=}64$ , mixed lr)	64	400K	19.36
Laplacian kernel ( $B{=}64$ )	64	100K	28.12

Appendix

Table 1: Comparison under different training budgets. We report the batch size, training steps, and FID for our method and prior baselines using a DiT-B/2 backbone for one-step latent-space generation.

Figure 7: Ablation study on feature-space optimization. We compare three settings: using the default feature set with thousands of features (top), using only four features (middle), and operating directly in the original space (bottom).

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Drifting models have recently gained attention for generating high-quality samples in a single forward pass. During training, they learn a push-forward map by following a vector-valued field, the drift field. We ask whether this procedure is equivalent to optimizing a scalar loss and find that, in general, it is not: drift fields are not conservative and cannot be written as the gradient of any scalar potential. We identify the position-dependent normalization as the source of non-conservatism, with the Gaussian kernel as the unique radial exception. Guided by this, we introduce the sharp kernel

k^\#

and a sharp-normalized drift field that is conservative for general radial kernels. The resulting vector field is the gradient of a scalar potential that can be optimized directly using stochastic gradient descent. Moreover, the field has the form of a score difference of kernel density estimates, and gives exact equilibrium identifiability. Thus, sharp normalization closes the gap to related literature, such as Wasserstein gradient-flows and denoising score matching, also for non-Gaussian kernels. Empirically, sharp normalization preserves the performance of the original drifting objective, suggesting that the non-conservative flexibility is not required for high-quality generation.

Abstract

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