Δ-flow

Δ-flowΔ Flow Matching for Robot Foundation Models

Learn the change from history, not from fixed Gaussian noise.

Kuangji Zuo* Tuo An* Xinying Guo Gen Li Mengfei Zhao Jiaqi Bai Bofan Lyu Zhuoyuan Yu Jinghan Yang Jiayi Li Jindou Jia†‡ Jianfei Yang†

MARS Lab, Nanyang Technological University

* Equal contribution‡ Project lead† Corresponding authors
arXiv soon PDF soon GitHub BibTeX
Overview: Δ-flow uses history-based sources for one-step generation. Training panels show 4–6× fewer epochs across the evaluated backbones and latency panels show 1.54–4.45× speedups over ten-step Gaussian sampling. On ImageWAM, boundary peaks in action change and jerk fall by 48% and 67%, respectively.
Overview. Δ-flow starts each source from recent history plus mild noise and learns only the change Δ.
6×fewer epochs for ImageWAM to reach 90% LIBERO success
1stepinference, up to 4.45× faster than ten-step Gaussian
88.55%ImageWAM success on LIBERO-Plus
4.5×lower ImageWAM action jerk than one-step Gaussian
Motivation

Why Start from Gaussian Noise?

Actions and scenes change continuously, and the robot already knows its recent past. Δ-flow starts the flow there, so the network only learns a small change Δ.

Gaussian source

Noise → future

  • Transports unrelated noise to the future
  • Iterative sampling; one-step sampling can reduce smoothness
Δ-flow

History + noise → future

  • Uses recent history as a temporal reference
  • Learns a residual for one-step generation
Interactive · illustration

Drag τ from noise (1) to action (0)

Illustrative trajectories. Grey: executed history. Dashed: the next action chunk.

Gaussian source—
Δ-flow source—
Method

One Recipe, Three Backbones

History-based sources. Existing backbones. One-step prediction.

Δ-flow for world-action models (left) and for vision-language-action models (right).
Architecture. WAMs learn visual and action changes; VLAs learn action changes. One Euler update per branch.
WAM · image

ImageWAM

Vision ΔAction Δ

FLUX.2-klein-base-4B + action DiT

WAM · video

Fast-WAM

Vision ΔAction Δ

Wan2.2-TI2V-5B + action DiT

VLA

π0.5

Vision cond.Action Δ

PaliGemma + Gemma action expert

Training

Up to 6× Fewer Training Epochs

ImageWAM reaches 90% LIBERO success in 1 epoch instead of 6; Fast-WAM reaches 95% in 2 instead of 8.

Success rate versus training epoch on LIBERO and LIBERO-Plus for ImageWAM, Fast-WAM and π0.5.
Training progress. LIBERO (top) and LIBERO-Plus (bottom). Fast-WAM reference: Gaussian sparse-joint. Error bars: SD across 40 tasks where available.
π0.5 with LoRA: one-step Δ-flow reaches 92.20% after one epoch, above the ten-step baseline's 91.30% after five.
π0.5 with LoRA. 92.20% after 1 epoch, exceeding the ten-step Gaussian baseline after 5 (91.30%).
Generalization

Stronger Generalization

ImageWAM reaches 88.55% on LIBERO-Plus, without extra embodied-data pretraining or additional fine-tuning.

(a) LIBERO
ModelEmb.PTSpatialObjectGoalLongOverall
VLA
OpenVLA✓84.788.479.253.776.5
UniVLA✓96.596.895.692.095.2
π0.5✓98.8096.8098.8094.0097.10
π0.5 (Δ-flow)✓99.60100.0099.6096.8099.00
WAM
LingBot-VA✓98.599.697.298.598.5
Motus✓96.899.896.697.697.7
ABot-M0.5✓100.099.899.498.499.4
Cosmos Policy×98.1100.098.297.698.5
ImageWAM×98.8099.2098.8095.2098.00
ImageWAM (Δ-flow)×99.2099.6098.4094.0097.80
Fast-WAM×————96.60
Fast-WAM-joint (Gaussian)×98.4099.4098.6093.4097.45
Fast-WAM-joint (Δ-flow)×98.00100.0098.6095.4098.00
(b) LIBERO-Plus
ModelEmb.PTCameraRobotLanguageLightBackgroundNoiseLayoutOverall
VLA
OpenVLA✓0.83.523.08.134.815.228.515.6
UniVLA✓1.846.269.669.081.021.231.942.9
π0.5✓66.0473.6181.3997.3791.3676.7086.7580.70
π0.5 (Δ-flow)✓73.3679.3583.8096.9497.3081.2687.3484.53
WAM
LingBot-VA✓40.983.086.482.353.164.476.269.5
Motus✓48.088.089.083.078.057.089.076.0
ABot-M0.5✓70.587.488.694.089.775.585.283.4
Cosmos Policy×75.863.381.796.588.992.782.282.2
ImageWAM×82.5552.9086.9997.3790.2492.0781.3182.49
ImageWAM (Δ-flow)×82.2473.9493.6298.6993.7793.5788.3988.55
Fast-WAM×44.864.270.289.057.375.073.467.24
Fast-WAM-joint (Gaussian)×58.2276.7195.7792.3864.6874.5881.1177.51
Fast-WAM-joint (Δ-flow)×70.0478.7785.4391.7768.1288.3880.7980.58
Success rate (%). Emb.PT: extra embodied-data pretraining. Bold: best per model group. Fast-WAM-joint uses matched sparse, one-step settings. LIBERO-Plus Overall is category-weighted; literature protocols vary.
Inference

One-Step, Smooth Inference

One-step generation is 1.54–4.45× faster than ten-step Gaussian sampling.

Latency and success rate for ImageWAM, Fast-WAM and π0.5.
Latency and success. Complete policy-call latency on A100-80GB. Dashed/solid lines: LIBERO/LIBERO-Plus. π0.5 uses LoRA; Fast-WAM uses sparse-joint inference.

ImageWAM achieves 4.5× lower mean action jerk than one-step Gaussian sampling.

Action change and jerk around chunk boundaries on ImageWAM.
Action smoothness. ImageWAM on LIBERO-Long. Boundary peaks fall 48% for action change and 67% for jerk versus one-step Gaussian sampling. Bands: paired-bootstrap 95% CI.
Interactive · illustration

What happens at chunk boundaries

Top: executed actions. Bottom: jerk. Shaded: each new chunk.

Mean jerk—
Boundary jump—
vs 10-step Gaussian—

Illustration only. Measured results are in the figure above.

No retraining

Action History Transfers to Frozen WAMs

Swap the action source in frozen ImageWAM and Fast-WAM policies. No retraining.

sourceGaussian noise
sourcePast actions + noise
frozen policy0 parameters updated
ImageWAM · action
97.50→98.10%
Fast-WAM · action
95.55→96.15%
LIBERO success at one step, frozen policy. LIBERO-Plus at one step: ImageWAM 81.65 → 82.82%; Fast-WAM 68.28 → 67.58%.
Full table
PolicySourceStepsLIBEROLIBERO-Plus
ImageWAMGaussian1098.0082.49
ImageWAMGaussian197.5081.65
ImageWAMPast action198.1082.82
Fast-WAMGaussian1096.6067.24
Fast-WAMGaussian195.5568.28
Fast-WAMPast action196.1567.58

Steps: inference sampling updates. Policy weights remain frozen.

Analysis

Ablations

Which parts of Δ-flow matter, measured on ImageWAM.

Both flows help

Original82.49
Action Δ85.82
Action + visual Δ88.55

ImageWAM · LIBERO-Plus · 50% image-latent prior.

Visual Endpoint Consistency Helps under Shift

Action + visual Δ87.37
w/o visual EC84.96

Removing visual EC lowers LIBERO-Plus success by 2.41 points. Uses the 80% image-latent prior.

Analysis

Design Lessons

What else matters when putting Δ-flow into a foundation model.

01 · Fast-WAM

Predict fewer future frames

Four future frames outperform eight over the same horizon: 72.90 → 78.09% on LIBERO-Plus.

Fast-WAM variants on LIBERO and LIBERO-Plus.
Fast-WAM variants. All models are jointly trained on the four LIBERO suites.
02 · ImageWAM

Stop action gradients into the world model

Stop-gradient improves future-frame PSNR (15.04 → 20.86 dB) and LIBERO-Plus success (83.24 → 84.96%), without visual EC.

ImageWAM with full gradient versus stop-gradient: LIBERO suites, future-frame PSNR, and LIBERO-Plus perturbations.
Action-gradient transfer. Black numbers: stop-gradient success. Blue: difference from full gradient in percentage points. Right: future-frame PSNR.
03 · ImageWAM

Endpoint loss alone almost suffices

EC-only action training nearly matches final performance; adding flow matching accelerates early learning.

Action objectives in ImageWAM: EC only versus FM plus EC.
Action objectives. 80% image-latent prior; visual objectives unchanged. EC: L1 endpoint consistency.
04 · ImageWAM

Use local delta actions

Local-delta actions generalize better overall: 85.82% versus 61.07% for 7D absolute actions on LIBERO-Plus.

Action representations on LIBERO and LIBERO-Plus.
Action representations. Black: local-delta success. Blue: gain over 7D absolute actions. LIBERO-Plus Overall is category-weighted.
Related work

Beyond A2A

Δ-flow extends our A2A flow matching.

Action space

No latent encoder needed

Vision too

History-based visual and action sources in WAMs

Foundation scale

VLAs and WAMs

New studies

Representation and coupling

Citation

BibTeX

Preprint details are pending. The citation will be updated when the paper is released.

@article{deltaflow,
  title   = {$\Delta$ Flow Matching for Robot Foundation Models},
  author  = {Zuo, Kuangji and An, Tuo and Guo, Xinying and Li, Gen and
             Zhao, Mengfei and Bai, Jiaqi and Lyu, Bofan and Yu, Zhuoyuan and
             Yang, Jinghan and Li, Jiayi and Jia, Jindou and Yang, Jianfei},
  journal = {arXiv preprint},
  year    = {2026}
}