Learning Conditional Source Distribution via Flow Reversal for Temporal Flow Matching

Kuan-Hsun Tu; Hsuan-Chi Liu; Jia-Wei Liao; Chien-Sheng Chiang; Cheng-Fu Chou; Tsung-Wei Ke

cs.CVOct 4, 2026

Learning Conditional Source Distribution via Flow Reversal for Temporal Flow Matching

Authors: Kuan-Hsun Tu, Hsuan-Chi Liu, Jia-Wei Liao, Chien-Sheng Chiang, Cheng-Fu Chou, Tsung-Wei Ke

Organizations: National Taiwan University

Abstract

We introduce CNP-Flow, a flow matching framework for temporal generation that learns conditional source distributions through flow reversal. Whereas standard conditional flow matching (FM) incorporates conditioning through the vector field and draws source samples from a standard Gaussian, CNP-Flow uses a conditional noise predictor (CNP) to produce an isotropic Gaussian source for each temporal condition. The CNP is supervised by source samples obtained through flow reversal, which maps observed targets backward through a pretrained FM model. A three-stage pipeline pretrains the FM model, trains the CNP, and fine-tunes the FM model using the learned source distribution, while preserving the FM backbone architecture. Across video prediction, video interpolation, and 7-DoF Franka robot motion planning, CNP-Flow consistently improves generation quality. It also matches baseline performance with fewer function evaluations. Project page: https://embodiedai-ntu.github.io/cnpflow

Figures & tables

Figure 1: CNP-Flow learns condition-dependent source distributions for temporal generation. CNP-Flow uses a Conditional Noise Predictor (CNP) to predict a source distribution for each condition. This source-side conditioning reduces FVD by up to $75\%$ over RIVER on video prediction, and yields more precise motion-planning trajectories with fewer collisions.

Figure 2: 2D toy example of temporal generation . Colors indicate temporal phase along the target circular trajectory. (a) Vanilla FM draws source samples from a standard Gaussian. Samples associated with neighboring temporal conditions can be widely separated, and the generated sequence deviates from the target trajectory. (b) CNP-Flow learns a conditional source distribution with a smoother arrangement of samples by temporal phase (see the zoomed region), and the generated sequence follows the target trajectory more closely. Detailed experimental settings are provided in Appendix F .

Figure 3: Overview of the proposed three-stage training pipeline. Stage 1: A pretrained FM is used to obtain flow-reversal-induced source samples, defining a source distribution aligned with the target data. Stage 2: The CNP is trained to model this distribution by predicting a conditional Gaussian prior via NLL and KL objectives. Stage 3: The FM model is fine-tuned using source samples drawn from the learned conditional prior, resulting in more coherent and target-aligned transport trajectories.

Dataset	$p\rightarrow k$	Method	PSNR $\uparrow$	SSIM $\uparrow$	FVD $\downarrow$
BAIR64	$1\rightarrow 15$	RIVER ( Davtyan et al., 2023 )	–	–	73.50
		MAGVIT ( Yu et al., 2023 )	19.30	0.787	62.00
		iVideoGPT ( Wu et al., 2024 )	20.40	0.823	75.00
		CAR-Flow ( Chen et al., 2025 )	20.08	0.836	71.37
		PYoCo ( Ge et al., 2023 )	20.13	0.837	61.88
		CSFM ( Kim et al., 2026 )	20.16	0.836	64.20

Table 1: Results on video prediction. CNP-Flow improves over the fixed-source RIVER baseline and remains competitive with recent source-learning methods. CNP-Flow results are averaged over three seeds.

Method	Success $\uparrow$	Valid (%) $\uparrow$	EE pos (cm) $\downarrow$	EE ori ( ∘ ) $\downarrow$	Collision $\downarrow$
MPD ( Carvalho et al., 2025 )	97%	86.6	1.9	1.2	0.0030
Fixed-source FM	96%	84.9	1.6	1.1	0.0038
CNP-Flow (Ours)	97%	88.5	0.8	0.5	0.0023

Table 3: Results on motion planning. All methods use a matched 15-step sampling budget, and results are averaged over three seeds.

Figure 4: Inference efficiency on KTH prediction. Curves show PSNR/SSIM averaged across generated samples per video. CNP-Flow reaches the standard Gaussian PSNR plateau (

\approx 26.5

) at NFE

=5

and converges higher (

\approx 28.2

) by NFE

=10

; SSIM shows a similar trend. The crossover point is highlighted on each panel.

Figure 5: Qualitative comparison. Left : CLEVRER interpolation. Fixed-source FM drifts at the highlighted frames, while CNP-Flow matches the ground truth. Right : Franka Panda motion planning. Each panel overlays 50 sampled trajectories (red: colliding, green: collision-free). CNP-Flow produces more collision-free plans than MPD and fixed-source FM.

Video prediction (KTH)
Source $\bm{x}_{0}$	PSNR $\uparrow$	SSIM $\uparrow$	FVD $\downarrow$
$\mathcal{N}(\bm{0},\mathbf{I})$	30.40	0.860	180.00
$\bm{\mu}_{\phi}(\mathcal{C}^{i})$	31.46	0.908	83.82
$\mathcal{N}(\bm{\mu}_{\phi}(\mathcal{C}^{i}),\bm{\sigma}_{\phi}^{2}(\mathcal{C}^{i})\mathbf{I})$	34.25	0.943	44.76

Table 4: Ablation studies. Left : source parameterization on KTH

10\!\rightarrow\!30

video prediction, isolating the contribution of the learned mean and learned variance. Right : source learning and backbone fine-tuning (FT) on Franka Panda motion planning. In both tasks, metrics improve as the source better matches the condition.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Fixed-source FM	CNP-Flow	Latency increase	Additional memory
KTH / BAIR64	207.1 ms/frame	214.2 ms/frame	$3.43\%$	${\sim}15$ MB
CLEVRER	211.9 ms/frame	219.1 ms/frame	$3.39\%$	${\sim}15$ MB

Appendix

Table 5: Inference overhead of CNP-Flow. Latency is measured over 1,000 generated frames under matched hardware, batch-size, and solver settings.

Component	KTH	CLEVRER	BAIR64
Flow reversal steps	100	20	50
FM fine-tune length	200k steps	20k steps	50k steps
FM fine-tune batch	64	128	64
FM fine-tune LR	$5{\times}10^{-5}$	$5{\times}10^{-5}$	$5{\times}10^{-5}$

Appendix

Table 6: Video prediction fixed-source FM and flow reversal configuration. Source samples obtained through flow reversal are cached only for CNP supervision and are not used at inference time.

Component	KTH	CLEVRER
FM pretrain steps	300k	300k
FM pretrain batch	32	16
Flow reversal steps	100	40
FM fine-tune steps	200k	200k
FM fine-tune batch	32	16
FM fine-tune LR	$1{\times}10^{-4}$	$1{\times}10^{-4}$

Appendix

Table 7: Video interpolation fixed-source FM and flow reversal configuration. Since no compatible interpolation checkpoint is available, the fixed-source CFM backbone is trained from scratch.

Component	Setting
Backbone	Convolutional encoder
Input	Conditional latents + temporal-distance embedding
Output	$\mu,\log\sigma^{2}$
Base channels	64
Channel multipliers	$(1,2,4,8)$
Residual blocks per resolution	2

Appendix

Table 8: Video CNP architecture. The same architecture is used for video prediction and video interpolation.

Task	Dataset	Length	Batch	KL warm-up
Prediction	KTH	50 epochs	64	10 epochs
Prediction	CLEVRER	15 epochs	1024	10 epochs
Prediction	BAIR64	15 epochs	64	5 epochs
Interpolation	KTH	200k steps	32	5 epochs
Interpolation	CLEVRER	200k steps	16	2 epochs

Appendix

Table 9: Video CNP training configuration. All video CNPs are trained with Adam using learning rate

1{\times}10^{-4}

and KL weight

\lambda_{\mathrm{KL}}=1{\times}10^{-4}

Dataset	PSNR $\uparrow$	SSIM $\uparrow$	FVD $\downarrow$
Video prediction
BAIR64 $1\!\rightarrow\!15$	$20.15\pm 0.13$	$0.8390\pm 0.0021$	$61.17\pm 1.58$
KTH $10\!\rightarrow\!30$	$34.25\pm 0.26$	$0.9432\pm 0.0025$	$44.76\pm 1.53$
KTH $10\!\rightarrow\!40$	$32.69\pm 0.22$	$0.9303\pm 0.0040$	$54.88\pm 3.85$
CLEVRER $2\!\rightarrow\!14$	$42.23\pm 0.19$	$0.9935\pm 0.0002$	$24.18\pm 2.46$
Video interpolation

Appendix

Table 10: Variability across runs for CNP-Flow. Results are reported as mean

\pm

standard deviation over three independent runs for video prediction and interpolation.

Component	Setting
Backbone	Residual MLP
Input	Start state + EE goal pose + context embedding
Raw condition dimension	26
Context embedding dimension	128
Output	$\mu,\log\sigma^{2}$
Hidden dimension	1024

Appendix

Table 11: Motion planning CNP architecture. The CNP predicts an isotropic Gaussian source distribution over the learnable B-spline control-point noise.

Method	Valid (%) $\uparrow$	EE pos. (cm) $\downarrow$	EE ori. ( ∘ ) $\downarrow$	Collision $\downarrow$
Diffusion prior	$64.3\pm 1.5$	$5.3\pm 0.1$	$7.5\pm 0.5$	$0.0402\pm 0.0013$
Fixed-source FM	$61.1\pm 3.8$	$3.8\pm 0.2$	$5.6\pm 0.5$	$0.0445\pm 0.0034$
CNP-Flow	$64.3\pm 4.6$	$\mathbf{2.1\pm 0.2}$	$\mathbf{2.9\pm 0.3}$	$\mathbf{0.0366\pm 0.0045}$

Appendix

Table 12: Motion-planning performance without post-hoc refinement. Results are reported as mean

\pm

standard deviation over three independent runs.

Regime	Task	Split	Method	MSE $\downarrow$	M-MSE $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$
Rigid	push_cube	IND	Vanilla FM	0.002886	0.017354	26.29	0.9546
		IND	CNP-Flow (Ours)	0.002192	0.013373	27.80	0.9647
		OOD	Vanilla FM	0.002977	0.016039	26.78	0.9539
		OOD	CNP-Flow (Ours)	0.002338	0.012326	28.13	0.9629
Deformable	push_rope	IND	Vanilla FM	0.000213	0.002708	36.75	0.9878
		IND	CNP-Flow (Ours)	0.000207	0.002608	36.88	0.9889

Appendix

Table 13: Results on the ACWM-Phys physical world-model benchmark at

240{\times}240

resolution. We evaluate one task from each physical regime on both in-distribution (IND) and out-of-distribution (OOD) splits. All results are averaged over three independent seeds.

Source, frozen FM	PSNR $\uparrow$	SSIM $\uparrow$	FVD $\downarrow$
Standard Gaussian	30.29	0.8591	179.97
CNP w/o flow reversal supervision	30.46	0.8612	179.54
CNP w/ flow reversal supervision	30.85	0.9008	100.91

Appendix

Table 14: Effect of flow reversal supervision on source learning. The Stage-1 FM is frozen for all variants.

Source	Pairwise LPIPS $\uparrow$
Standard Gaussian	0.0666
CNP-Flow	0.0686

Appendix

Table 15: Prediction diversity on BAIR64. We report the average pairwise LPIPS among predictions generated from the same condition.

Dataset	Round 1	Round 2	Round 3
BAIR64	61.17	60.48	67.07
KTH	44.76	35.04	36.37

Appendix

Table 16: Iterative Stage-2/Stage-3 refinement. FVD is reported after each round of source learning and FM refinement.

Flow reversal solver	FVD $\downarrow$
Dopri	$61.73\pm 1.47$
Heun-100	$\mathbf{61.16\pm 0.99}$
Euler-100	$61.17\pm 1.56$

Appendix

Table 17: Robustness to the numerical solver used for flow reversal on BAIR64 prediction. Values are mean

\pm

standard deviation over three seeds.

Method	$\\|\bm{x}_{0}-\bm{x}_{1}\\|_{2}$ $\downarrow$	Path length $\downarrow$	Turning (rad) $\downarrow$	Endpoint dev. $\downarrow$
Standard Gaussian	19.1715	18.6780	0.8292	2.7362
CNP-Flow (Ours)	17.4091	17.0088	0.7586	2.4710

Appendix

Table 18: Transport geometry on BAIR64. Both methods are evaluated on the same 256 condition–target pairs using Euler-100.

Coupling	Distance $\downarrow$	Path $\downarrow$	Turning $\downarrow$	Endpoint dev. $\downarrow$
Flow reversal pair	15.7342	15.7185	0.2261	0.4913
Conditional OT	17.3715	16.2187	1.0610	3.9173
CNP-Flow (Ours)	17.4091	17.0088	0.7586	2.4710

Appendix

Table 19: Alternative Stage-3 couplings on BAIR64 prediction. Left : transport geometry on 256 condition–target pairs using Euler-100. Right : downstream generation quality. Direct pairing via flow reversal requires the target and is therefore reported only as a training-time geometry reference.

Panel of Figure 6 — Figure 6: Toy example visualization of the flow reversal mapping and transport distance analysis.

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In recent years Flow Matching has become a prominent method for generative modeling robot motion generation. In its generic form Flow Matching is an ODE-based neural sampler that is trained by regressing empirical flow fields associated with motion samples as data. However, in robot motion generation we often have additional constraints that might not be present in the collected data. The majority of current approaches train the flow on the available data and use inference-time guidance to enforce task-specific constraints. To address this mismatch, we propose \textbf{ConFlow}, a constraint-guided flow matching framework that incorporates constraint information directly into the training objective via differentiable barrier or cost functions. To address design specifications such as smoothness and boundary conditions, we propose replacing the standard Gaussian source distribution used in flow matching training with a conditional Gaussian Process. Our approach also uses infeasible demonstrations as negative supervision, improving constraint satisfaction without requiring additional expert data. Experiments on a two-robot navigation task demonstrate that ConFlow achieves lower collision rates and higher trajectory quality than standard flow matching baselines, with or without inference-time guidance. These results validate training-time constraint integration as an effective approach to closing the training--inference gap in generative motion models.

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