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 π0.5 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, π0.5 exceeds its baseline's five-epoch performance after one epoch. Single-step inference gives 1.54×, 2.49×, and 4.45× speedups on π0.5, ImageWAM, and Fast-WAM, respectively. It preserves action smoothness, achieving a 4.5× 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: Δ -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, Δ -flow constructs each source from its corresponding recent history with mild Gaussian perturbation and learns only the residual change Δ . 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: Δ -flow for robot foundation policies. History-based sources combine past representations with Gaussian noise, so the model learns past-to-future changes. Left: Δ -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: Δ -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 π0.5 . 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 Δ -flow, and blue bars denote complete Δ -flow. The π0.5 panels compare one-step and ten-step Gaussian sampling with one-step action Δ -flow. Dashed lines track Δ -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 π0.5 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 Δ -flow. Annotations give percentage-point differences.
Model
Emb.PT
Spatial
Object
Goal
Long
Overall
VLA
OpenVLA ( Kim et al., 2025 )
✓
84.7
88.4
79.2
53.7
76.5
UniVLA ( Bu et al., 2025 )
✓
96.5
96.8
95.6
92.0
95.2
π0.5 ( Black et al., 2025a )
✓
98.80
96.80
98.80
94.00
97.10
π0.5 ( Δ -flow)
✓
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 Δ -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 π0.5 . 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 Δ -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 Δ -flow, and one-step complete Δ -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
–
Δ -flow, action only
98.40
85.82
+3.33
Δ -flow, action + visual
97.80
88.55
+6.06
Table 3: Contributions of the action and visual Δ -flows on ImageWAM. Change (pp) is the signed LIBERO-Plus difference from the original Gaussian -source model.
Configuration
LIBERO
LIBERO-Plus
Δ -flow, action only
98.10
85.75
Δ -flow, action + visual (80% image-latent prior)
97.70
87.37
w/o LECv
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
π0.5 (Full)
π0.5 (LoRA)
Training hardware
8× A100
8× A100
8× A100
8× 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
5×10−5
5×10−5
Warmup
5% of steps
5% of steps
100 steps
100 steps
Appendix
Table 5: Training hyperparameters for ImageWAM, Fast-WAM-joint, and π0.5 . Full and LoRA denote full-parameter and LoRA fine-tuning, respectively. Batch size is global.
Setting
ImageWAM ( Δ -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 Δ -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
π0.5 ( Δ -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 π0.5 with action Δ -flow.
Method
Steps
Mean action jerk ↓
Gaussian
10
0.04221
Gaussian
1
0.19399
Δ -flow, action only
1
0.04185
Δ -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.
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
Md Tanvir Islam, Sai Navaneet Peddapalli, Sangmoon Lee +1
School of Electronic and Electrical Engineering, Kyungpook National University, Daegu 41566, Republic of Korea
World-action models (WAMs) transfer visual and motion priors from pretrained video generators to robot control by jointly modeling visual dynamics and actions. Existing WAMs, however, predict dense future frames during training, repeatedly modeling largely unchanged content and coupling action-conditioned dynamics to nuisance appearance variations. At inference, processing each complete observation with the heavy video expert bottlenecks few-step action generation. Accordingly, we propose DeltaWAM, which jointly predicts visual deltas and actions using dense-anchor, sparse-delta, and action streams, with three architectures that differ in representation and computation sharing. We further develop Streaming Delta Memory (SDM), which updates cached anchor context with compact observed deltas, reducing heavy video-expert processing. On RoboTwin, DeltaWAM with SDM improves average success over Fast-WAM from 81.3% to 85.4% in the clean setting and from 75.8% to 83.9% under visual randomization. The three architectures reduce training FLOPs by 17.78-23.77%, while SDM reduces one-step inference latency and FLOPs by 36.57% and 31.55%, respectively; real-world evaluations further show the highest overall success rate and normalized progress among the evaluated policies. Code: https://github.com/AIGeeksGroup/DeltaWAM. Website: https://aigeeksgroup.github.io/DeltaWAM.
Han Yan, Zishang Xiang, Haokai Jiang +6
School of Computer Science, Peking University · AI2 Robotics
Flow matching has recently become a new standard for behavior cloning in robotic manipulation. However, state-of-the-art flow matching policies suffer from a systematic structural mismatch: they rely on a globally fixed isotropic source distribution despite the strongly fragmented and heteroscedastic structure of robotic action spaces. This agnostic initialization forces the model to learn highly entangled vector fields, bottlenecking training efficiency and limiting overall policy performance. To address this limitation, we introduce Latent Action Guided Flow Matching (LAFM), a novel framework that replaces the monolithic Gaussian with an adaptive library of learned prior distributions. By grounding these distributions using a latent action model, LAFM maps current observations to discrete motion primitives, selecting a specialized base distribution that provides an informed, structurally aligned initialization for the denoising process. This dynamic adaptivity naturally accommodates heteroscedasticity in human demonstrations and makes transport trajectories shorter and less entangled. Empirically, LAFM substantially outperforms standard flow matching formulations, increasing task success rates by 23.4% in real-world robotic deployments and by 10.4% on the LIBERO-90 benchmark. Furthermore, we demonstrate that LAFM achieves state-of-the-art results, surpassing massively pre-trained vision-language-action models while utilizing significantly smaller architectures.
Bruno Machado, Alexandre Chapin, Emmanuel Dellandrea +1