$Δ$ Flow Matching for Robot Foundation Models

Kuangji Zuo; Tuo An; Xinying Guo; Gen Li; Mengfei Zhao; Jiaqi Bai; Bofan Lyu; Zhuoyuan Yu

cs.ROOct 11, 2026

$Δ$ Flow Matching for Robot Foundation Models

Authors: Kuangji Zuo, Tuo An, Xinying Guo, Gen Li, Mengfei Zhao, Jiaqi Bai, Bofan Lyu, Zhuoyuan Yu, +4 more

Organizations: MARS Lab, Nanyang Technological University

Abstract

Flow matching is a common generative recipe for robot foundation models. Vision-language-action models (VLAs) generate action chunks, while world-action models (WAMs) additionally generate future visual states. Both start generation from a fixed Gaussian prior, despite temporal continuity between past and future states and actions. This requires costly training and iterative sampling, especially for WAMs with an additional visual branch. We introduce $Δ$ -flow, which constructs the source from recent history with mild noise rather than from a fixed Gaussian prior. This moves the source closer to the target, reducing transport to a small residual $Δ$ that can be learned efficiently and enables single-step inference. We integrate $Δ$ -flow into the VLA policy $π_{0.5}$ and the WAM policies ImageWAM and Fast-WAM. On LIBERO, ImageWAM reaches 90% success in one epoch rather than six, and Fast-WAM reaches 95% in two epochs rather than eight. With LoRA fine-tuning, $π_{0.5}$ exceeds its baseline's five-epoch performance after one epoch. Single-step inference gives $1.54\times$ , $2.49\times$ , and $4.45\times$ speedups on $π_{0.5}$ , ImageWAM, and Fast-WAM, respectively. It preserves action smoothness, achieving a $4.5\times$ reduction in action jerk compared with one-step Gaussian sampling. $Δ$ -flow reaches state-of-the-art generalization performance on LIBERO-Plus. Action-history sources can be applied to pretrained WAM policies without retraining. Replacing the Gaussian action source at inference largely preserves task success with a single sampling step and can slightly improve it. We further study how action representation and visual information in action learning affect history-to-future prediction in $Δ$ -flow. These results show that $Δ$ -flow is simple yet effective, improving the efficiency and generalization of robot foundation models.

Figures & tables

Figure 1: $\mathbf{\Delta}$ -flow matching learns past-to-future changes for efficient robot foundation models. Unlike Gaussian -source flow matching, which reconstructs future visual states and actions from noise through iterative sampling, $\mathbf{\Delta}$ -flow constructs each source from its corresponding recent history with mild Gaussian perturbation and learns only the residual change $\mathbf{\Delta}$ . This shorter transport substantially improves training efficiency, enables fast single-step inference, strengthens generalization under distribution shift, and preserves smooth action generation across chunk boundaries.

Figure 2: $\mathbf{\Delta}$ -flow for robot foundation policies. History-based sources combine past representations with Gaussian noise, so the model learns past-to-future changes. Left: $\Delta$ -flow for world-action models. The model learns both visual and action changes from history to future targets. The action expert uses intermediate visual representations, with stop-gradient blocking action-loss gradients from updating the visual branch during training. Right: $\Delta$ -flow for vision-language-action models. The model learns action changes from executed history to future actions, using the original vision-language conditioning. At inference, a single Euler update transforms each history-based source into its future prediction.

Figure 3: Training progress on LIBERO and LIBERO-Plus. Success rates are shown over training epochs for ImageWAM, Fast-WAM, and $\pi_{0.5}$ . The top panels show LIBERO and the bottom panels show LIBERO-Plus, evaluated without additional fine-tuning. White bars denote the original models, gray bars denote the Gaussian sparse-joint Fast-WAM baseline, light-blue bars denote action-only $\Delta$ -flow, and blue bars denote complete $\Delta$ -flow. The $\pi_{0.5}$ panels compare one-step and ten-step Gaussian sampling with one-step action $\Delta$ -flow. Dashed lines track $\Delta$ -flow, and annotations report percentage-point differences. Insets show later training epochs, and stars mark the highest evaluated success rate. Error bars show one standard deviation across the 40 base tasks.

Figure 4: Training efficiency on

\pi_{0.5}

with LoRA fine-tuning. The left panel shows LIBERO success over five training epochs. The right panel shows success across the four LIBERO suites at epoch five. The curves and bars compare the ten-step Gaussian-source baseline with one-step action

\Delta

-flow. Annotations give percentage-point differences.

VLA
Model	Emb.PT	Spatial	Object	Goal	Long	Overall
OpenVLA ( Kim et al., 2025 )	$\checkmark$	84.7	88.4	79.2	53.7	76.5
UniVLA ( Bu et al., 2025 )	$\checkmark$	96.5	96.8	95.6	92.0	95.2
$\pi_{0.5}$ ( Black et al., 2025a )	$\checkmark$	98.80	96.80	98.80	94.00	97.10
$\pi_{0.5}$ ( $\mathbf{\Delta}$ -flow)	$\checkmark$	99.60	100.00	99.60	96.80	99.00
WAM

Table 1: Success rates on LIBERO and LIBERO-Plus. Success rates are reported in %. Emb.PT indicates additional embodied-data pretraining, excluding general-purpose vision, language, or video backbone pretraining. Bold marks the highest value in each column within the VLA and WAM groups separately. Literature results retain their original reporting protocols. For models evaluated by us, each

\mathbf{\Delta}

-flow variant and its original counterpart use the same hardware and matched evaluation settings. Both Fast-WAM-joint rows use the sparse joint configuration with one-step sampling.

Figure 5: Inference latency and task success on ImageWAM, Fast-WAM, and

\pi_{0.5}

. Bars show mean complete local policy-call latency on an A100 GPU, including online text encoding and visual source construction, with values read from the left axes. White, gray, and blue bars denote the original models, Gaussian sparse-joint Fast-WAM, and

\mathbf{\Delta}

-flow, respectively. Dashed lines with square markers show LIBERO success, and solid lines with circle markers show LIBERO-Plus success, with values read from the right axis.

Figure 6: Action continuity at chunk boundaries on ImageWAM. The panels show mean action change (left) and action jerk (right) around transitions between consecutive action chunks. The four curves compare Gaussian sampling with ten or one step, one-step action-only

\mathbf{\Delta}

-flow, and one-step complete

\mathbf{\Delta}

-flow. All four methods are evaluated on the same LIBERO-Long tasks with identical initial states. Each policy predicts a 16-action chunk and executes the first 12 actions before replanning. Step 0 is the first action of the new chunk. Background shading indicates the new chunk, and bands around the curves show paired-bootstrap 95% confidence intervals. Arrows annotate relative differences in boundary peaks.

Policy	Source	Steps	LIBERO	LIBERO-Plus
ImageWAM	Gaussian	10	98.00	82.49
ImageWAM	Gaussian	1	97.50	81.65
ImageWAM	Past action	1	98.10	82.82
Fast-WAM	Gaussian	10	96.60	67.24
Fast-WAM	Gaussian	1	95.55	68.28
Fast-WAM	Past action	1	96.15	67.58

Table 2: Training-free action-source adaptation on LIBERO and LIBERO-Plus. Only the inference-time source of a frozen policy is changed, without updating model parameters. Success rates are in %. Steps denotes the number of inference sampling steps.

Configuration	LIBERO	LIBERO-Plus	Change (pp)
Original ImageWAM	98.00	82.49	–
$\mathbf{\Delta}$ -flow, action only	98.40	85.82	$+3.33$
$\mathbf{\Delta}$ -flow, action $+$ visual	97.80	88.55	$\mathbf{+6.06}$

Table 3: Contributions of the action and visual

\mathbf{\Delta}

-flows on ImageWAM. Change (pp) is the signed LIBERO-Plus difference from the original Gaussian -source model.

Configuration	LIBERO	LIBERO-Plus
$\mathbf{\Delta}$ -flow, action only	98.10	85.75
$\mathbf{\Delta}$ -flow, action $+$ visual (80% image-latent prior)	97.70	87.37
w/o $\mathcal{L}_{\mathrm{EC}}^{v}$	98.60	84.96

Table 4: Endpoint consistency on ImageWAM. Success rates (%) are reported for the final trained models. The action-only model provides a reference. The visual endpoint-consistency ablation uses the 80% image-latent prior.

Figure 7: Success rates (%) of Fast-WAM variants on LIBERO and LIBERO-Plus. All models are jointly trained on the four LIBERO suites.

Figure 8: Action-gradient transfer into the visual branch. Comparison across LIBERO suites, future-image prediction PSNR, and LIBERO-Plus perturbations. Black numbers in the success-rate panels report stop-gradient success rates. Blue annotations show differences from full gradient in percentage points.

Figure 9: Action objectives in ImageWAM (success rate, %). Both models use the 80% image-latent prior. The action FM/EC weights are

1.0/0.5

for FM+EC and

0/1.0

for EC only; the visual objectives are unchanged. EC is L1 endpoint consistency.

Figure 10: Action representations on LIBERO and LIBERO-Plus. We compare 7D local delta actions with axis-angle rotations, 7D absolute actions with axis-angle rotations, and 10D absolute actions with 6D rotations. The left panel shows success rates across LIBERO suites, and the right panel shows success rates across LIBERO-Plus perturbation categories. Black numbers report local-delta success. Blue annotations show local delta minus 7D absolute in percentage points.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	ImageWAM	Fast-WAM-joint	$\pi_{0.5}$ (Full)	$\pi_{0.5}$ (LoRA)
Training hardware	$8\times$ A100	$8\times$ A100	$8\times$ A100	$8\times$ A100
Batch size	80	128	384	384
Training epochs	10	10	10	5
Optimizer	AdamW	AdamW	AdamW	AdamW
Peak learning rate	$10^{-4}$	$10^{-4}$	$5\times 10^{-5}$	$5\times 10^{-5}$
Warmup	5% of steps	5% of steps	100 steps	100 steps

Appendix

Table 5: Training hyperparameters for ImageWAM, Fast-WAM-joint, and

\pi_{0.5}

. Full and LoRA denote full-parameter and LoRA fine-tuning, respectively. Batch size is global.

Setting	ImageWAM ( $\mathbf{\Delta}$ -flow)
Prediction horizon	16 steps
Action chunk length	16 commands
Action history length	16 commands
Executed actions before replanning	12 commands
Future frame offset	$16$
Number of future frames	1

Appendix

Table 6: Temporal layout and inference settings of ImageWAM with

\mathbf{\Delta}

-flow. Future-frame offsets are measured in environment steps relative to decision time

t

Setting	Joint-Dense	Joint-Sparse (Sparse8)
Prediction horizon	32 steps	32 steps
Action chunk length	32 commands	32 commands
Visual sampling stride $r$	4	8
Future frame offsets	$\{4,8,12,16,20,24,28,32\}$	$\{8,16,24,32\}$
Number of future frames	8	4
Clip frames including the current frame	9	5

Appendix

Table 7: Temporal layouts of the dense and sparse Fast-WAM-Joint configurations. Offsets are measured in environment steps relative to decision time

t

. Future-frame counts exclude the current conditioning frame.

Setting	$\pi_{0.5}$ ( $\mathbf{\Delta}$ -flow)
Prediction horizon	10 steps
Action chunk length	10 commands
Action history length	10 commands
Executed actions before replanning	5 commands
Action dimensions	7
Action sampling steps	1

Appendix

Table 8: Action-chunk and inference settings of

\pi_{0.5}

with action

\mathbf{\Delta}

-flow.

Method	Steps	Mean action jerk $\downarrow$
Gaussian	10	0.04221
Gaussian	1	0.19399
$\mathbf{\Delta}$ -flow, action only	1	0.04185
$\mathbf{\Delta}$ -flow, action + visual	1	0.04335

Appendix

Table 9: Trajectory-averaged action jerk on ImageWAM. Results are averaged over 20 matched successful trajectories from all ten LIBERO-Long tasks, using the first six action-command dimensions and excluding the gripper. Lower is better.

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Generative vision-language-action policies have advanced robot manipulation, but they often exhibit instability under noise, partial observability, and stochastic initial conditions. During extended rollouts, small velocity errors accumulate, degrading execution reliability. Existing diffusion and flow-based policies typically assume homoscedastic residuals and lack explicit uncertainty modeling within action generation, limiting robustness during iterative rollout. We propose SUREFlow, a state-space uncertainty-aware residual flow matching framework built on a Mamba backbone. The method jointly predicts action velocities and input-dependent residual uncertainty, enabling selective refinement of unreliable action dimensions without environment feedback while preserving computational efficiency. On LIBERO, SUREFlow achieves 92.5% average success rate (SR), outperforming the Mamba-based MaIL by 34.2%. On LIBERO-PRO, it attains around 49% SR using only 179M parameters, achieving performance comparable to large VLAs with 3-7B parameters. SUREFlow source code is available on: https://github.com/tanvirnwu/SUREFlow

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