Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective

Jian-Feng Cai; Zhengyi Su; Chao Wang

cs.LGSep 7, 2026

Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective

Authors: Jian-Feng Cai, Zhengyi Su, Chao Wang

Organizations: Department of Mathematics The Hong Kong University of Science and Technology Hong Kong SAR, China · Department of Statistics and Data Science Southern University of Science and Technology Shenzhen, Guangdong, China

Abstract

Flow matching, together with classifier-free guidance (CFG), is widely used in generative modeling, yet much of the theoretical understanding remains distribution-wise. Since practical sampling follows individual trajectories, distribution-level guarantees alone do not fully capture how trajectories interact with the data geometry or how guidance reshapes it. To overcome this limitation, we establish a unified stagewise geometric theory of attraction and absorption for both continuous dynamics and explicit Euler discretization. Specifically, with $t\in[0,1]$ running from noise to data, we show that unconditional flow trajectories are successively attracted toward a neighborhood of the global mean, the data convex hull, and a neighborhood of a possibly nonconvex local cluster. Across these stages, the corresponding distance satisfies a common contraction estimate, yielding an ${O}(1-t)$ decay of the distance in the final stage. For CFG, the same structure persists with an extrapolated mean, an inflated conditional convex hull, and, near the target cluster, the restored local geometry of conditional flow matching. We further show that a general time schedule $a(t)$ replaces the $O(1-t)$ decay by $O(1-a(t))$ . Together, these results provide a unified particle-level geometric account of flow matching and CFG across continuous and discrete sampling.

Figures & tables

Stage	Ref	Dynamics	Geometry and result	Contraction / terminal rate
Early	Theorem 4	Uncond. flow (cont.)	mean-ball attraction	$(1-t)/(1-s)$
	Theorem 5	Uncond. flow (disc.)	mean-ball attraction	$(1-t_{i})/(1-t_{j})$
	Theorem 12	CFG (cont.)	extrapolated mean-ball attraction	$(1-t)/(1-s)$
	Theorem 38	CFG (disc.)	extrapolated mean-ball attraction	$(1-t_{i})/(1-t_{j})$
Early and inter.	Theorem 6	Uncond. flow (cont.)	$\operatorname{Conv}(\mathcal{D})$ attraction	$(1-t)/(1-s)$
	Theorem 7	Uncond. flow (disc.)	$\operatorname{Conv}(\mathcal{D})$ attraction	$(1-t_{i})/(1-t_{j})$

Table 1: Stagewise summary of the main theoretical results.

Figure 1: Schematic illustration of the geometry behind UFM. The smoothed squared-distance term

\operatorname{dist}_{t}^{2}(\bm{x},t\mathcal{D})

captures the moving data geometry associated with

t\mathcal{D}

, while the full objective

f_{t}(\bm{x})=\frac{1}{2}\operatorname{dist}_{t}^{2}(\bm{x},t\mathcal{D})-\frac{1-t}{2}\|\bm{x}\|^{2}

determines the ideal flow direction.

Panel of Figure 2 — Figure 2: Stagewise moving geometries for the unconditional trajectory. In the early stage, $\bm{x}_{t}$ moves toward the moving mean ball $tB_{r_{\mathrm{mean}}}(\bar{\bm{y}})$ . The scaled data convex hull $t\operatorname{Conv}(\mathcal{D})$ provides a global localization geometry. In the final stage near a target cluster $\Omega$ , $\bm{x}_{t}$ moves toward the cluster neighborhood $tB_{r_{\Omega}}(\Omega)$ .

Panel of Figure 3 — Figure 3: Conditions behind the final-stage theorem. (i) The working neighborhood $B_{R}(\Omega)$ lies inside the $\rho_{\Omega}$ -neighborhood on which the metric projection is single-valued. (ii) The polynomial lower mass bound holds uniformly at every $\bm{y}\in\Omega$ . (iii) The separation from $B_{R}(\Omega)$ to the rest of the data is larger than $R+D_{\Omega}$ , equivalently $\gamma_{\Omega}=\delta_{\Omega}^{2}-(R+D_{\Omega})^{2}>0$ .

Panel of Figure 4 — Figure 4: Stagewise moving geometries under classifier-free guidance. In the early stage, guidance changes the attractor to the extrapolated mean $\bar{\bm{y}}_{\mathrm{cfg}}$ . The neighborhood of the scaled, inflated conditional convex hull provides a coarse localization geometry; after capture and separation, this neighborhood is refined to a smaller one. In the final stage near a conditional cluster $\Omega_{c}$ , $\bm{x}_{t}$ moves toward $tB_{\tilde{r}_{\Omega_{c}}}(\Omega_{c})$ .

Panel of Figure 5 — Figure 5: Convex-hull attraction and absorption. Left: the Euler trajectory and selected scaled convex-hull snapshots; the star marks the first entry into $t_{i}\operatorname{Conv}(\mathcal{D})$ , and the dashed curves show the moving convex hull at selected grid times. Right: $\operatorname{dist}(\bm{z}_{i},t_{i}\operatorname{Conv}(\mathcal{D}))$ decreases to zero and remains zero after the first entry.

Panel of Figure 6 — Figure 6: Final-stage local-cluster attraction and absorption. Left: five Euler trajectories enter their corresponding scaled cluster neighborhoods; stars mark the first entry, squares mark the final positions, and dashed contours show the initial, first-entry, and final scaled-neighborhood snapshots. Right: $\operatorname{dist}(\bm{z}_{i}^{j},t_{i}B_{r}(\Omega_{j}))$ decreases to zero and remains zero after first entry.

Panel of Figure 7 — Figure 7: CFG trajectory geometry. Left: larger $w$ shifts the extrapolated mean $\bar{\bm{y}}_{\mathrm{cfg}}$ and redirects the early trajectory, while the full trajectories subsequently turn toward $\Omega_{5}$ ; dashed rays indicate scaled extrapolated means, circles mark trajectory snapshots, and squares mark scaled-mean snapshots. Right: In the final stage, $\operatorname{dist}(\bm{z}_{i},t_{i}B_{0.05}(\Omega_{5}))$ decreases to zero and remains zero after first entry.

Panel of Figure 8 — Figure 8: Early-stage CFG attraction to the extrapolated mean with radius $r=0.05$ . Left: $w=1.5$ . Right: $w=3.0$ . Here, $\operatorname{dist}(\bm{z}_{i},t_{i}B_{r}(\bar{\bm{y}}_{\mathrm{cfg}}))$ decreases and stays below the linear reference $(1-t_{i})\|\bm{x}_{0}\|$ .

Panel of Figure 9 — Figure 9: Final-stage Prediction-gap decay. The measured gaps are well fit by the asymptotic form $\frac{C_{1}}{1-t_{i}}\exp(-\frac{C_{2}}{(1-t_{i})^{2}})$ for $w=1.5$ and $w=3.0$ ; the shaded interval indicates the grid-time window used for the fit.

Panel of Figure 10 — Figure 10: ImageNet-256 goldfinch early-attractor visualization. Top: unconditional, class, and extrapolated means. Bottom: samples from the same initialization. Larger $w$ amplifies class-specific structure, while very large $w$ over-saturates the output.

Panel of Figure 11 — Figure 11: Late-stage removal of CFG extrapolation on ImageNet-256 with $w=2$ and 20 steps. After the indicated step, we remove $w(\bm{v}_{c}-\bm{v}_{\emptyset})$ and finish with $\bm{v}_{\emptyset}$ ; removing too early weakens performance, while removing after 12 or 16 steps stays close to the full CFG.

Switch-off	after 6	after 12	after 16	full CFG
NFE	26	32	36	40
FID $\downarrow$	8.45	6.24	6.23	6.24

Table 2: FID

\downarrow

for late-stage CFG switch-off on ImageNet-256 using DiT-XL/2-256,

w=2

, 20 steps, step size

1/20

, and 50k generated images.

Scheme A
Scheme B
Scheme C
Scheme A
Scheme B
Scheme C
Scheme A

Figure 12: Intermediate denoising states for Schemes A, B, and C under Euler sampling with fixed base step size

\Delta t=1/105

. The three blocks show daisy, cup, and horse trajectories, respectively; each row contains 15 frames.

Dataset	NFE	$w$	A: standard	B: sampling reparam.	C: training reparam.
MNIST	60	1.5	10.24	10.67	9.78
CIFAR-10	100	1.1	5.20	5.38	4.85
ImageNet $32\times 32$	100	1.2	5.05	5.45	4.93

Table 3: Ablation of training and sampling reparameterization. FID

\downarrow

for (A) standard training and Euler sampling, (B) standard training with reparameterization in sampling, and (C) direct general-schedule training with Euler sampling.

Figure 13: DOPRI5 adaptive-solver cost by class for the directly trained models corresponding to Schemes A and C. Relative to $a(t)=t$ , the nonlinear schedule $a(t)=t+t^{2}-t^{3}$ reduces both mean NFE/image and mean wall-clock time/image across all reported classes.

Schedule	FID $\downarrow$	Mean NFE/image $\downarrow$	Mean time/image (s) $\downarrow$
$a(t)=t$	5.74	156.21	10.67
$a(t)=t+t^{2}-t^{3}$	5.67	120.84	8.19

Table 4: Quality in FID(

\downarrow

) and sampling cost under DOPRI5. Both rows use the same DOPRI5 solver tolerances (

\mathrm{atol}=\mathrm{rtol}=10^{-5}

) and guidance scale

w=3.5

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 14 — Figure 14: Appendix-proof dependency map. The appendix is read from top to bottom.

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Classifier-free guidance (CFG) improves conditional generation in Flow Matching, but strong guidance can distort the generated distribution and reduce diversity. We provide a geometric account of this behavior by viewing Flow Matching as a time-varying gradient flow and characterizing how CFG reshapes its underlying potential. This view explains how stronger alignment can be accompanied by mean displacement and trajectory concentration, and motivates controlling guidance through the model-implied terminal posterior mean. We therefore propose Posterior-Mean-Capped CFG (PMC-CFG), a training-free, per-sample method that adaptively retains the strongest feasible guidance without additional network evaluations. Experiments on synthetic and large-scale image-generation benchmarks show that PMC-CFG limits guidance-induced distortion and concentration while improving the alignment--diversity trade-off, with particularly strong benefits when nominal guidance is large.

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