Flow Matching for Robot Foundation Models

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.