FastOPD: On-Policy-Destillation für leichtgewichtige VLA-Bereitstellung

Abstract

Vision-Language-Action (VLA) foundation models have scaled rapidly to enhance manipulation performance and generalizability, but this scaling incurs high computational costs that render real-world deployment increasingly challenging. Existing approaches typically mitigate this issue by designing smaller architectures or reducing the iterative denoising steps in flow-based policies. In this work, we propose FastOPD, a foundation-to-lightweight VLA framework that enables the practical deployment of large-scale VLAs through efficient on-policy distillation. Specifically, FastOPD adapts a flow map for single-state teacher supervision and combines it with a self-consistency objective to construct a compact student that learns the teacher dynamics. Furthermore, we theoretically demonstrate that minimizing this objective allows the distilled student to recover a distribution on par with that induced by an ideal few-step teacher model. We evaluate FastOPD across diverse foundation policies in simulation and real-world experiments. On LIBERO, FastOPD retains 84% of the performance of \pi_{0.5} with only two inference steps, reducing inference latency by 78.1% while outperforming existing few-step distillation baselines in average success rate. With LingBot-VLA as the teacher, FastOPD improves the single-step success rate over the base student by 15.9 percentage points on RoboTwin 2.0. We further demonstrate its applicability to a World Action Model (WAM) and deploy a compact student distilled from MolmoAct2 on a real robot.

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1 Introduction

Recent progress in robot learning has moved toward large-scale foundation policies, ranging from Vision-Language-Action (VLA) models (Collaboration, 2024; Black et al., 2025b; Black et al., 2025a; Bjorck et al., 2025; Wu et al., 2026; Fang et al., 2026a) to emerging World Action Models (WAMs) (Kim et al., 2026; Ye et al., 2026; Yuan et al., 2026). In particular, recent methods such as DreamZero (Ye et al., 2026) and Cosmos Policy (Kim et al., 2026) leverage pretrained video generation models to capture rich visual dynamics and achieve strong generalization and manipulation performance. However, these powerful foundation priors also lead to increasingly large policy backbones and high computational cost, creating a tension between model capability and deployment efficiency on physical robots. Lightweight VLA models such as SmolVLA (Shukor et al., 2025) address the efficiency side of this trade-off by directly designing smaller policies with less than 0.5B parameters.

A complementary direction is to transfer the capability of a powerful foundation policy into a compact student through knowledge distillation. Knowledge Distillation (KD) (Hinton et al., 2015; Mansourian et al., 2025) has been widely studied for transferring the capability of a large teacher model into a smaller student. Specifically, On-Policy Distillation (OPD) (Agarwal et al., 2024) supervises the student using teacher predictions on its own generated trajectories, reducing distribution mismatch and compounding errors from offline distillation. OPD also provides dense supervision along the student trajectory, in contrast to the sparse rewards commonly used in reinforcement learning. While OPD has been actively explored in LLM post-training, recent works have extended it to diffusion and flow models (Li et al., 2026a; Fang et al., 2026b). However, directly applying OPD to flow-based policies is computationally inefficient. It requires rolling out the student’s denoising trajectory and repeatedly evaluating the teacher at the visited timesteps, resulting in high training cost for large foundation teachers.

In this work, we revisit few-step flow-map models through the lens of on-policy distillation (OPD) and introduce FastOPD, a foundation-to-lightweight VLA framework for efficient policy distillation. Our goal is to transfer the capabilities of a large-scale VLA into a compact student while substantially reducing both training and inference costs, without sacrificing policy performance (see Fig. 1). To reduce the training cost of conventional OPD, FastOPD combines on-policy teacher supervision of the student’s velocity field with a self-consistency objective (Frans et al., 2025). Rather than querying teacher supervision throughout an entire generation trajectory, the self-consistency objective propagates local velocity supervision to finite-interval shortcut transitions, as illustrated in Fig. 2. This allows the student to approximate the teacher dynamics without explicitly constructing or evaluating a teacher flow map over each shortcut transition. We further show theoretically that FastOPD implicitly recovers the distribution induced by the teacher’s ideal shortcut model, providing a distributional justification for few-step distillation.

Importantly, the same self-consistency formulation naturally induces an any-step flow map (Boffi et al., 2024; Song et al., 2023; Kim et al., 2024; Geng et al., 2025; Sabour et al., 2025), allowing the distilled student to operate with substantially fewer sampling steps at inference. FastOPD therefore compresses the foundation VLA along two complementary dimensions: model capacity and sampling complexity, enabling deployment on resource-constrained, consumer-grade hardware. The proposed formulation also enables stable training in the challenging few-step regime without adversarial objectives or costly higher-order derivative computations. A concurrent work, VLA-OPD (Zhong et al., 2026), also studies on-policy distillation for VLA models, but primarily uses expert supervision to improve policy performance. In contrast, FastOPD focuses on transferring a foundation policy into a lightweight few-step student.

We evaluate FastOPD on two simulation benchmarks and one real-world manipulation task, using diverse foundation teachers with 3B–6B parameters (Black et al., 2025a; Yuan et al., 2026; Fang et al., 2026a). To maximize both architectural and inference efficiency, we adopt SmolVLA as the lightweight student policy. We compare against representative few-step distillation methods, including CTM, DMD, and iMF. Across these settings, FastOPD enables the compact student to recover up to 84% of the performance of teacher models that are more than 7.3\times larger, using only one or two sampling steps. This substantially reduces inference cost while retaining a large fraction of the foundation policy’s performance. To the best of our knowledge, FastOPD is the first OPD framework for VLAs that jointly reduces both model size and the number of sampling steps while retaining the capabilities of a large-scale foundation policy.

2 Background

Flow-based Models. Diffusion and flow-based models (Song et al., 2021; Ho et al., 2020; Esser et al., 2024; Lipman et al., 2023) generate high-quality samples through iterative denoising. Let p_{1}=\mathcal{N}(\mathbf{0},\mathbf{I}) be a Gaussian prior and p_{0} the context-conditioned data distribution over actions. Flow matching (Lipman et al., 2023; Esser et al., 2024) defines a stochastic interpolant {\bm{x}}_{t}=(1-t){\bm{x}}_{0}+t{\bm{x}}_{1}, with {\bm{x}}_{0}\sim p_{0}, {\bm{x}}_{1}\sim p_{1},\ t\in[0,1], and learns the marginal velocity field {\bm{v}}({\bm{x}},t):=\mathbb{E}[{\bm{x}}_{1}-{\bm{x}}_{0}\mid{\bm{x}}_{t}={\bm{x}}] via

\mathbb{E}_{t,{\bm{x}}_{0},{\bm{x}}_{1}}\big[\|{\bm{v}}_{\theta}({\bm{x}}_{t},t)-({\bm{x}}_{1}-{\bm{x}}_{0})\|_{2}^{2}\big].

Sampling integrates the probability-flow ODE \dot{{\bm{x}}}_{t}={\bm{v}}({\bm{x}}_{t},t) from t=1 to t=0 , for example, with Euler steps, {\bm{x}}_{t_{i}}={\bm{x}}_{t_{i-1}}+{\bm{v}}({\bm{x}}_{t_{i-1}},t_{i-1})\,\Delta t where \Delta t:=t_{i}-t_{i-1}. While these models capture high-quality multimodal action distributions, accurate integration typically requires many denoising steps, motivating few-step distillation for faster inference.

Vision-Language-Action Models. Vision-Language-Action (VLA) models (Collaboration, 2024; Black et al., 2025b; Black et al., 2025a; Bjorck et al., 2025; Wu et al., 2026; Liu et al., 2024; Octo Model Team et al., 2024) couple a pretrained vision-language backbone with an action decoding head to map an observation and a language instruction to a distribution over robot actions, most commonly via a flow or diffusion action expert (Chi et al., 2023; Black et al., 2025b) trained with flow matching loss. Recent foundation VLA models illustrate the scale required for strong multi-task generalization, typically exceeding 2B parameters and trained on over 10,000 hours of multimodal robot data spanning multiple embodiments (Black et al., 2025b; Black et al., 2025a; Wu et al., 2026), but remain expensive to deploy. Several works (Octo Model Team et al., 2024; Shukor et al., 2025; Wen et al., 2025; Reuss et al., 2025) instead train smaller VLAs; SmolVLA (Shukor et al., 2025), for example, reduces model size to 450 million parameters by using a small VLM and layer skipping. However, it still requires N{=}10 denoising steps per action chunk and does not benefit from the capabilities of larger foundation policies.

Flow Map Models. A wide range of few-step generative methods aim to reduce iterative sampling cost. Distillation-based approaches (Sabour et al., 2025; Boffi et al., 2024), including Consistency Models (CM) (Song et al., 2023), and Consistency Trajectory Models (CTM) (Kim et al., 2024), distill a pre-trained multi-step teacher into a few-step student, while score-distillation methods directly match teacher and student distributions (Yin et al., 2024b; Yin et al., 2024a). Distillation-free approaches, such as MeanFlow (MF) (Geng et al., 2025) and Shortcut Models (Frans et al., 2025), instead train from scratch using self-consistency constraints. A flow map model {\bm{f}}_{\theta}({\bm{x}},s,t) parameterizes the state-to-state transition over an arbitrary interval [s,t]\subset[0,1], so that a single jump {\bm{f}}_{\theta}({\bm{x}}_{s},s,t)={\bm{x}}_{s}+(t-s){\bm{u}}_{\theta}({\bm{x}}_{s},s,t) at the interpolant {\bm{x}}_{s}=(1-s){\bm{x}}_{0}+s{\bm{x}}_{1} replaces an entire sequence of Euler substeps. For example, Frans et al. (2025) combines flow matching loss with the self-consistency constraint at the midpoint time m=({s+t})/{2}:

\mathcal{L}(\theta)=\mathbb{E}_{s,t,{\bm{x}}_{1},{\bm{x}}_{0}}\left[||{\bm{v}}_{\theta}({\bm{x}}_{s},s)-({\bm{x}}_{1}-{\bm{x}}_{0})||_{2}^{2}+||{\bm{f}}_{\theta}({\bm{x}}_{s},s,t)-{\bm{f}}_{\theta^{-}}({\bm{f}}_{\theta^{-}}({\bm{x}}_{s},s,m),m,t)||_{2}^{2}\right]

The first term enforces standard flow matching, while the second encourages the direct transition from s to t to agree with two shorter transitions composed through m. In robot manipulation, these ideas have been adapted to accelerate diffusion- and flow-based policies such as Diffusion Policy (Chi et al., 2023). Consistency Policy (CP) (Prasad et al., 2024) adapts CTM (Kim et al., 2024); OneDP (Wang et al., 2025) and SDM Policy (Jia et al., 2024) use distribution matching for one-step action generation; and recent methods learn average velocity fields (Geng et al., 2025; Geng et al., 2026; Sheng et al., 2026; Guo et al., 2026) or shortcut objectives (Frans et al., 2025; Luan et al., 2026; Yan et al., 2025; Li et al., 2026b) regularized by flow matching. For instance, SnapFlow (Luan et al., 2026) constructs its target from two Euler steps but trains only on fixed endpoints, while OFP (Li et al., 2026b) derives its target from nested sub-intervals with a self-guided regularizer. Despite these advances, few-step policies often suffer from training instability and degraded quality as the number of denoising steps decreases. This trade-off is particularly critical in robot manipulation, where low latency must be achieved without sacrificing the precision and stability required for physical execution, and still remains insufficiently addressed (Shukor et al., 2025).

On-policy Distillation. Given a pretrained teacher diffusion model, on-policy distillation (OPD) supervises the student to match the teacher’s one-step transition kernel p_{T}(\cdot\mid{\bm{x}}_{t_{i}}) along the student’s own rollout {\bm{x}}_{0:N}\sim p_{S,\theta}, by minimizing the reverse KL objective (Li et al., 2026a). Since the student and teacher share the same diffusion coefficient, resulting in identical covariance, its per-step KL reduces to a closed-form squared distance between the means, \mathrm{KL}(p_{S}\|p_{T})\propto\|\mu_{S}({\bm{x}}_{t_{i}};\theta)-\mu_{T}({\bm{x}}_{t_{i}})\|_{2}^{2}. Despite providing dense supervision, this objective necessitates evaluating the teacher at every single denoising step, resulting in computational inefficiency. FastOPD circumvents this training bottleneck by supervising the student at a single on-policy state per trajectory. Formulated within a few-step flow map regime, the distilled policy is naturally equipped for real-time robot control.

3 FastOPD: Fast On-Policy Distillation

Overview. Our goal is to efficiently distill a large pretrained teacher {\bm{v}}_{\phi} into a compact flow map student {\bm{u}}_{\theta}, while achieving performance on par with the teacher. Specifically, we parameterize the flow map as {\bm{f}}_{\theta}({\bm{x}},s,t)={\bm{x}}+(t-s){\bm{u}}_{\theta}({\bm{x}},s,t), which satisfies the boundary condition {\bm{f}}_{\theta}({\bm{x}},s,s)={\bm{x}}. We denote its diagonal by {\bm{v}}_{\theta}({\bm{x}},t):={\bm{u}}_{\theta}({\bm{x}},t,t), which corresponds to the instantaneous velocity. As illustrated in Fig. 3, we embed s and t independently via sinusoidal positional encoding and fuse them through an additional projection layer \psi(\cdot). Here, we omit the condition c for brevity.

The key idea of FastOPD is to match the teacher only at sampled on-policy states and propagate this local velocity supervision to finite-step flow maps through self-consistency. Accordingly, FastOPD jointly optimizes on-policy velocity matching and finite-interval self-consistency:

\displaystyle\mathcal{L}(\theta)=\underbrace{\mathbb{E}\Big[\big\|{\bm{u}}_{\theta}({\bm{x}}_{s},s,t)-\tfrac{1}{2}\big(\mathrm{sg}[{\bm{v}}_{\theta}({\bm{x}}_{s},s)]+\mathrm{sg}[{\bm{v}}_{\theta}(\hat{{\bm{x}}}_{m},m)]\big)\big\|^{2}\Big]}_{\text{finite-interval self-consistency}}+\lambda\underbrace{\mathbb{E}\Big[\big\|{\bm{v}}_{\theta}(\tilde{{\bm{x}}}_{t},t)-{\bm{v}}_{\phi}(\tilde{{\bm{x}}}_{t},t)\big\|^{2}\Big]}_{\text{on-policy flow map distillation}}.

Here, \mathrm{sg}[\cdot] denotes the stop-gradient operator, and \tilde{{\bm{x}}}_{t} denotes a sampled on-policy state, while \hat{{\bm{x}}}_{m} denotes the midpoint state induced by the student. We next describe how the two terms implement local teacher matching and its propagation to finite-step flow maps.

On-policy Flow Map Distillation. We first consider how to efficiently transfer the teacher dynamics to the student. Conventional flow-based OPD evaluates the teacher at every denoising step along the student trajectory, resulting in substantial computational overhead. Instead, we supervise the student at a single sampled on-policy state per trajectory (see Fig. 3(b)). Specifically, we obtain the state with a single flow-map transition from noise, \tilde{{\bm{x}}}_{t}=\mathrm{sg}\big[{\bm{f}}_{\theta}({\bm{x}}_{1},1,t)\big], which induces the on-policy marginal \tilde{{\bm{x}}}_{t}\sim p_{\theta}. We then match the teacher and student transitions only at this sampled timestep t. Let p_{\theta}(\cdot\mid\tilde{{\bm{x}}}_{t}) and p_{\phi}(\cdot\mid\tilde{{\bm{x}}}_{t}) denote the corresponding one-step transition kernels of the student and teacher, respectively. We formulate the single-state transition matching objective as

\mathbb{E}_{t\sim p(t)}\mathbb{E}_{\tilde{{\bm{x}}}_{t}\sim p_{\theta}}\left[\text{KL}\left(p_{\theta}(\cdot\mid\tilde{{\bm{x}}}_{t})\|p_{\phi}(\cdot\mid\tilde{{\bm{x}}}_{t})\right)\right].

For a flow map, the instantaneous velocity on the diagonal coincides with the underlying velocity field under the tangent condition (Boffi et al., 2025) 1. Using this identity, the reverse KL in Eq. (4) can be expressed directly in terms of the student and teacher velocity fields, as shown in the following proposition.

Proposition 1 (On-policy Flow Map Distillation).

Let \sigma_{t}>0 be a diffusion coefficient. For any t\in(0,1) and step \Delta t<0, the Euler–Maruyama kernels induced by {\bm{u}}_{\theta}(\cdot,t,t) and {\bm{v}}_{\phi}(\cdot,t) satisfy

D_{\mathrm{KL}}\left(p_{\theta}(\cdot\mid\tilde{{\bm{x}}}_{t})\,\middle\|\,p_{\phi}(\cdot\mid\tilde{{\bm{x}}}_{t})\right)=w_{\sigma}(t,\Delta t)\left\|{\bm{u}}_{\theta}(\tilde{{\bm{x}}}_{t},t,t)-{\bm{v}}_{\phi}(\tilde{{\bm{x}}}_{t},t)\right\|_{2}^{2},

where w_{\sigma}(t,\Delta t)=\frac{|\Delta t|}{2\sigma_{t}^{2}}\left(1+\frac{\sigma_{t}^{2}(1-t)}{2t}\right)^{2}.

Proposition 1 reduces the single-state transition matching objective to a weighted velocity regression. We therefore define the practical on-policy flow map distillation (OPFD) objective as

\mathcal{L}_{\text{OPFD}}(\theta)=\mathbb{E}_{c,t\sim p(t)}\left[\mathbb{E}_{\tilde{{\bm{x}}}_{t}\sim p_{\theta}}\left[\|{\bm{v}}_{\theta}(\tilde{{\bm{x}}}_{t},t)-{\bm{v}}_{\phi}(\tilde{{\bm{x}}}_{t},t)\|^{2}_{2}\right]\right].

Eq. (6) can be viewed as the pointwise version of the original distribution matching in Eq. (4), since it is characterized by a deterministic flow ODE. We set the weighting function w_{\sigma}(t,\Delta t) to 1 in practice, following Li et al. (2026a). Full derivation is provided in Appendix A.

Self-consistency Property. OPFD transfers the teacher velocity to the student’s diagonal flow map {\bm{u}}_{\theta}({\bm{x}},t,t), but our goal is to learn a flow map over arbitrary finite interval (s,t) at the same time. We therefore propagate this local teacher supervision to finite-step transitions through self-consistency (see Fig. 3(c)). Following Shortcut Models (Frans et al., 2025), a direct transition from s to t should agree with two consecutive shorter transitions through the midpoint m=(s+t)/2. Unlike the original shortcut formulation, we construct the target using the instantaneous velocities at the diagonals of the two sub-intervals, {\bm{v}}_{\theta}({\bm{x}}_{s},s) and v_{\theta}(\hat{{\bm{x}}}_{m},m), as follows:

\mathcal{L}_{\text{SC}}(\theta)=\mathbb{E}_{c,s,t,{\bm{x}}}\left[\left\|{\bm{u}}_{\theta}({\bm{x}}_{s},s,t)-\frac{1}{2}\left[\mathrm{sg}\big[{\bm{v}}_{\theta}({\bm{x}}_{s},s)\big]+\mathrm{sg}\big[{\bm{v}}_{\theta}(\hat{{\bm{x}}}_{m},m)\big]\right]\right\|^{2}\right]

where \hat{{\bm{x}}}_{m}:={\bm{x}}_{s}+(m-s){\bm{v}}_{\theta}({\bm{x}}_{s},s).

The resulting target is anchored to teacher-supervised quantities at every interval. Note that the diagonal terms coincide with the quantities directly supervised by OPFD. Accordingly, we define the training objective as

\mathcal{L}_{\text{FastOPD}}=\mathcal{L}_{\text{SC}}+\lambda\mathcal{L}_{\text{OPFD}}.

In the following theorem, we demonstrate that the student implicitly recovers the teacher-induced flow map. Let {\bm{v}}_{\phi}({\bm{x}},t) be the teacher velocity field, and we define the teacher-induced ideal shortcut as

{{\bm{u}}_{\phi}}({\bm{x}},s,t)=\tfrac{1}{2}\big[{\bm{v}}_{\phi}({\bm{x}},s)+{\bm{v}}_{\phi}({\bar{{\bm{x}}}}_{m},m)\big],\quad\bar{{\bm{x}}}_{m}={\bm{x}}+(m-s)\,{\bm{v}}_{\phi}({\bm{x}},s),

with {{\bm{u}}}_{\phi}({\bm{x}},t,t)={\bm{v}}_{\phi}({\bm{x}},t). We then define the corresponding ideal flow map as {\bm{f}}_{\phi}({\bm{x}},s,t)={\bm{x}}+(t-s){\bm{u}}_{\phi}({\bm{x}},s,t).

Proposition 2.

Let q_{\theta} and \rho_{\theta} be the sampling distributions of \mathcal{L}_{\text{SC}} and \mathcal{L}_{\text{OPFD}}. Assume that {\bm{v}}_{\phi}(\cdot,t) is L-Lipschitz and that (s,{\bm{x}}_{s}) and (m,\hat{{\bm{x}}}_{m}) under q_{\theta} have density ratio at most \kappa with respect to \rho_{\theta}. Then,

D_{q_{\theta}}({\bm{u}}_{\theta},{\bm{u}}_{\phi})\leq C_{\lambda}\mathcal{L}_{\text{FastOPD}},

where D_{q_{\theta}}({\bm{u}}_{\theta},{\bm{u}}_{\phi}):=\mathbb{E}_{(s,t,{\bm{x}})\sim q_{\theta}}\|{\bm{u}}_{\theta}({\bm{x}},s,t)-{\bm{u}}_{\phi}({\bm{x}},s,t)\|_{2}^{2} and C_{\lambda}=\max\Big\{4,\ \frac{\kappa(2+L^{2}/4)}{\lambda}\Big\}.

Thus, OPFD transfers the teacher’s instantaneous velocity to the student, while self-consistency propagates this agreement from the diagonal velocity to finite-step shortcut maps.

Theorem 1.

Let p_{\theta} and p_{\phi} denote the terminal distributions induced by sampling with {\bm{f}}_{\theta} and {\bm{f}}_{\phi}, respectively, from the same initial noise on a fixed sampling schedule 1=t_{0}>t_{1}>\cdots>t_{K}=0, and let {\bm{x}}^{\theta}_{t_{k}} denote the intermediate states of the student. Suppose that the assumptions of Proposition 2 hold and there exists finite \kappa_{\mathrm{roll}}>0 such that, for every k,

D_{k}:=\mathbb{E}\big\|{\bm{u}}_{\theta}({\bm{x}}^{\theta}_{t_{k}},t_{k},t_{k+1})-{\bm{u}}_{\phi}({\bm{x}}^{\theta}_{t_{k}},t_{k},t_{k+1})\big\|_{2}^{2}\leq\kappa_{\mathrm{roll}}\,D_{q_{\theta}}({\bm{u}}_{\theta},{\bm{u}}_{\phi}).

Then,

W_{2}(p_{\theta},p_{\phi})\leq C\sqrt{\mathcal{L}_{\text{FastOPD}}},

where C>0 depends only on L, \kappa, \kappa_{\mathrm{roll}}, \lambda, and the sampling schedule.

Consequently, W_{2}(p_{\theta},p_{\phi})\to 0 as \mathcal{L}_{\text{FastOPD}}\to 0. Since {\bm{u}}_{\phi} is built entirely from the pretrained teacher’s velocity field, p_{\phi} is the few-step distribution induced by the teacher-induced flow map on the same sampling schedule. The theorem therefore shows that FastOPD allows the compact student to implicitly recover this distribution without explicitly constructing a shortcut teacher. The detailed algorithm for FastOPD is presented in Algorithm 1.

4 Experiments

Our experiments span two multitask simulation benchmarks (Sec. 4.1) and one real-robot task (Sec. 4.2). We freeze the vision–language backbone and fine-tune only the action expert and projections with the additional time projection layer described in Sec. 3. We evaluate each method using 1, 2, 4, and 10 denoising steps and compare it against the teacher and the initial student with 50 trials per task. Full experimental details are provided in Appendix B.

4.1 Simulation Experiments

Benchmarks. We evaluate FastOPD on LIBERO (Liu et al., 2023) and RoboTwin 2.0 (Chen et al., 2025). LIBERO evaluates single-arm manipulation across four standard evaluation suites—Spatial, Object, Goal, and Long—comprising 40 tasks in total. RoboTwin 2.0 (Chen et al., 2025) comprises 50 challenging bimanual manipulation tasks, with 2,500 demonstrations collected in clean scenes and 25,000 demonstrations collected under heavy scene randomization. We report results in both clean and randomized settings on unseen environments.

Baselines. For simulation experiments, we compare our method with three representative approaches to few-step generation: Consistency Trajectory Model (CTM) (Kim et al., 2024), Distribution Matching Distillation (DMD) (Yin et al., 2024b; Yin et al., 2024a), and improved Mean Flow (iMF) (Geng et al., 2026). Since iMF was originally formulated for training from scratch, we adapt it to the distillation setting by replacing its self-referential instantaneous velocity target with the teacher velocity. Except for DMD, a one-step generator, all baselines are designed as the flow-map architecture.

Teacher and Student Models. We use SmolVLA (Shukor et al., 2025) as the student model on both benchmarks, paired with \pi_{0.5} (Black et al., 2025a) on LIBERO, LingBot-VLA (Wu et al., 2026) and Fast-WAM (Yuan et al., 2026) on RoboTwin 2.0. All models are initialized from public benchmark-specific checkpoints.

LIBERO. In Tab. 3 and Tab. 3, we report the average success rates and inference latency on the LIBERO benchmark. Our method outperforms all few-step baselines and the initial student model at every denoising step in average success rate. As shown in Fig. 4, we observe a consistent performance improvement across all denoising steps, including at 10 steps. FastOPD further achieves the best success rates of 81.8% with only 2 sampling steps. In particular, our method shows better performance than the 10-step SmolVLA baseline with only a single step, while reducing inference latency from 193 ms to 50 ms. Together, the results demonstrate that FastOPD substantially improves the accuracy–latency trade-off of the compact student. We examine the contributions of each loss component to these accuracy gains in Sec. 4.3.

RoboTwin 2.0. Tab. 3 and Tab. 3 summarize the results. Notably, our method consistently achieves performance gains in the few-step regime, improving the single-step success rate of the base student from 35.3% to 51.2%, and outperforms all few-step distillation baselines at every sampling step. It does so at a latency comparable to SmolVLA, while being 10.0\times faster than the LingBot-VLA teacher. At 4 and 10 steps, FastOPD performs slightly below the base student, indicating that its benefit lies mainly in the few-step regime where the base student degrades.

Distilling a World Action Model. Recent World Action Models (WAMs) build on pretrained video generation models and have emerged as an alternative to foundation VLAs (Kim et al., 2026; Ye et al., 2026; Yuan et al., 2026). To examine whether FastOPD extends beyond VLA teachers, we distill Fast-WAM (Yuan et al., 2026), a 6B-parameter video-based teacher, into the same 451M SmolVLA student with an identical training recipe. As shown in Tab. 3, the student distilled from Fast-WAM achieves a 50.5% average success rate at a single denoising step, 15.2 percentage points over the initial SmolVLA baseline (35.3%). Furthermore, it reaches 56.3% at two steps, successfully transferring knowledge despite the substantial differences in architecture and pretraining between the teacher and the student. This demonstrates that FastOPD is a versatile approach that applies not only to VLAs but also to WAMs, since it only requires the teacher’s velocity field on the student’s own action trajectories, regardless of how the teacher is built.

4.2 Real-World Experiments

To validate its real-world applicability, we further design a real-world robot scenario, pnp-plate, in which the robot picks up the ball and places it on the plate. Experiments use the right arm of a bimanual YAM robot with a parallel-jaw gripper and top-view and wrist-mounted cameras that provide RGB observations. We train MolmoAct2 (Fang et al., 2026a), an action reasoning model that is suitable for real-world deployment, on our collected demonstrations for 14,000 iterations and use the resulting policy as the teacher. We initialize the student from SmolVLA, warm it up on the collected demonstrations with the standard flow matching objective (i.e., behavior cloning) for 10,000 iterations, and subsequently distill the teacher into the student using FastOPD. Full experimental details are provided in Appendix B.5.

Fig. 6 and Fig. 6 summarize the results. With four denoising steps, FastOPD achieves a 50% success rate, improving by 8 percentage points over the 42% of the base SmolVLA at the same number of steps. Notably, FastOPD shows faster and more accurate performance than the 10-step base student, improving the success rate from 44% to 50% while cutting the execution time from 19.32 s to 17.38 s. These results demonstrate that FastOPD can transfer the capability of a larger teacher to a compact student deployed on a physical robot with few steps.

4.3 Ablation Study

Training efficiency. FastOPD reduces training overhead by supervising the student at a single sampled on-policy state rather than across all denoising steps. As illustrated in Fig. 1, standard OPD (Li et al., 2026a) takes 9.51 hours per 1,000 training iterations while FastOPD requires only 1.43 hours, achieving a 5.7\times speedup. This translates directly into accelerated convergence (Fig. 1). FastOPD achieves a 78.2% 1-step success rate in just 10 hours, whereas conventional OPD requires roughly 57 hours to reach a similar level (79.1%). Consequently, FastOPD matches the performance of conventional OPD at a fraction of the computational cost.

Student Initialization. We study whether FastOPD can also start from a student that is initialized in a fully on-policy manner. To do so, we initialize the student using 5,000 iterations of conventional per-step OPD (Li et al., 2026a) instead of flow-matching behavior cloning. As shown in Tab. 4, the OPD-initialized student degrades severely at 4 steps (44.2%), likely due to the fixed 10-step discretization in OPD, whereas FastOPD recovers this case and improves all denoising steps. This indicates that OPD is a viable alternative to flow-matching-based behavior cloning for the initialization, although it incurs a substantially higher training cost, as discussed above (Fig. 1). It also shows that FastOPD is not tied to a specific initialization and can be applied on top of students distilled by other methods.

Loss components. Tab. 5 shows that the two objectives are complementary. Without teacher anchoring, SC alone collapses below 18% and OPFD alone fails to capture off-diagonal flow dynamics, degrading few-step generation. Their combination reaches 81.8% with two steps. While Luan et al. (2026) also anchors a fixed two-step Euler target with flow matching, FastOPD anchors to the teacher via OPFD and enforces consistency over arbitrary intervals, yielding an any-step flow map that inherits the teacher’s dynamics (see Appendix C).

Hyperparameters. The proposed method introduces a single hyperparameter \lambda, which controls the strength of the OPFD term. To study its effect, we sweep the OPFD weight \lambda\in\{1,0.5,0.1,0.01\} on the OPD-initialized student (Tab. 6). Performance collapses at \lambda=0.01, indicating that sufficient OPFD supervision is needed to anchor the self-consistency term, and remains stable within [0.1,0.5] with a mild trade-off between few- and multi-step performance.

Action execution steps. The policy executes the first K actions of each predicted chunk before replanning, trading responsiveness for inference overhead. Short execution horizons improve responsiveness through more frequent replanning, while longer horizons reduce inference overhead at the cost of extended open-loop execution. As shown in Fig. 7, we observe that both the teacher \pi_{0.5} and the student perform best at K=10, suggesting that the student inherits the teacher’s preferred replanning frequency.

5 Conclusion

In this paper, we present FastOPD, an efficient on-policy distillation framework that transfers large-scale VLAs into lightweight student models to maximize computational efficiency without compromising performance. Specifically, FastOPD streamlines the iterative denoising process by anchoring a single on-policy state to the teacher, coupled with a self-consistency objective for finite-interval transitions. In simulation and real-robot experiments across diverse VLA and WAM teachers, FastOPD improves over the base student in the few-step regime and outperforms existing few-step distillation methods, while substantially reducing training and inference cost. We hope this framework facilitates scalable, real-time deployment of foundation policies on physical robots.

References

1. Agarwal et al. (2024) R. Agarwal, N. Vieillard, Y. Zhou, P. Stanczyk, S. Ramos Garea, M. Geist, and O. Bachem On-policy distillation of language models: learning from self-generated mistakes. In International Conference on Learning Representations,
2. Bjorck et al. (2025) J. Bjorck, F. Castañeda, N. Cherniadev, X. Da, R. Ding, L. Fan, Y. Fang, D. Fox, F. Hu, S. Huang, et al. GR00T N1: an open foundation model for generalist humanoid robots. arXiv preprint arXiv:2503.14734.
3. Black et al. (2025a) K. Black, N. Brown, J. Darpinian, K. Dhabalia, D. Driess, A. Esmail, M. R. Equi, C. Finn, N. Fusai, et al. \pi_{0.5}: a vision-language-action model with open-world generalization. In Proceedings of the 9th Conference on Robot Learning, Proceedings of Machine Learning Research, Vol. 305, pp. 17–40.
4. Black et al. (2025b) K. Black, N. Brown, D. Driess, A. Esmail, M. R. Equi, C. Finn, N. Fusai, L. Groom, K. Hausman, B. Ichter, et al. \pi_{0}: a vision-language-action flow model for general robot control. In Robotics: Science and Systems,
5. Boffi et al. (2025) N. Boffi, M. Albergo, and E. Vanden-Eijnden How to build a consistency model: learning flow maps via self-distillation. Advances in Neural Information Processing Systems 38, pp. 37559–37595.
6. Boffi et al. (2024) N. M. Boffi, M. S. Albergo, and E. Vanden-Eijnden Flow map matching with stochastic interpolants: a mathematical framework for consistency models. arXiv preprint arXiv:2406.07507.
7. Cadene et al. (2024) R. Cadene, S. Alibert, A. Soare, Q. Gallouedec, A. Zouitine, S. Palma, P. Kooijmans, M. Aractingi, M. Shukor, D. Aubakirova, M. Russi, F. Capuano, C. Pascal, J. Choghari, K. Meftah, M. Ellerbach, J. Moss, and T. Wolf LeRobot: state-of-the-art machine learning for real-world robotics in Pytorch. Note: https://github.com/huggingface/lerobot
8. Chen et al. (2025) T. Chen, Z. Chen, B. Chen, Z. Cai, Y. Liu, Z. Li, Q. Liang, X. Lin, Y. Ge, Z. Gu, et al. RoboTwin 2.0: a scalable data generator and benchmark with strong domain randomization for robust bimanual robotic manipulation. arXiv preprint arXiv:2506.18088.
9. Chi et al. (2023) C. Chi, S. Feng, Y. Du, Z. Xu, E. Cousineau, B. Burchfiel, and S. Song Diffusion policy: visuomotor policy learning via action diffusion. In Robotics: Science and Systems,
10. Collaboration (2024) O. X. Collaboration Open X-Embodiment: robotic learning datasets and RT-X models. In International Conference on Robotics and Automation (ICRA),
11. Esser et al. (2024) P. Esser, S. Kulal, A. Blattmann, R. Entezari, J. Müller, H. Saini, Y. Levi, D. Lorenz, A. Sauer, F. Boesel, D. Podell, T. Dockhorn, Z. English, and R. Rombach Scaling rectified flow transformers for high-resolution image synthesis. In Proceedings of the 41st International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 235, pp. 12606–12633.
12. Fang et al. (2026a) H. Fang, J. Duan, D. Clay, S. Wang, S. Liu, W. Huang, X. Fan, W. Tsai, S. Chen, Y. R. Wang, S. Xing, J. Cho, J. S. Park, A. Eftekhar, P. Sushko, K. Farley, A. Wadhwa, C. Harrison, W. Han, Y. Lee, E. VanderBilt, R. Hendrix, S. Ellawela, L. Ngoo, J. Chai, Z. Ren, A. Farhadi, D. Fox, and R. Krishna MolmoAct2: action reasoning models for real-world deployment. arXiv preprint arXiv:2605.02881.
13. Fang et al. (2026b) Z. Fang, W. Huang, Y. Zeng, Y. Zhao, S. Chen, K. Feng, Y. Lin, L. Chen, Z. Chen, S. Cao, et al. Flow-OPD: on-policy distillation for flow matching models. arXiv preprint arXiv:2605.08063.
14. Frans et al. (2025) K. Frans, D. Hafner, S. Levine, and P. Abbeel One step diffusion via shortcut models. In International Conference on Learning Representations,
15. Geng et al. (2025) Z. Geng, M. Deng, X. Bai, J. Z. Kolter, and K. He Mean flows for one-step generative modeling. In Advances in Neural Information Processing Systems, Vol. 38.
16. Geng et al. (2026) Z. Geng, Y. Lu, Z. Wu, E. Shechtman, J. Z. Kolter, and K. He Improved mean flows: on the challenges of fastforward generative models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 30467–30476.
17. Guo et al. (2026) Y. Guo, W. Chen, and J. Zhang ReactVLA: fast and lightweight reactive robot manipulation via improved mean flow action generation. arXiv preprint arXiv:2606.14255.
18. Hinton et al. (2015) G. Hinton, O. Vinyals, and J. Dean Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531.
19. Ho et al. (2020) J. Ho, A. Jain, and P. Abbeel Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, Vol. 33, pp. 6840–6851.
20. Jia et al. (2024) B. Jia, P. Ding, C. Cui, M. Sun, P. Qian, S. Huang, Z. Fan, and D. Wang Score and distribution matching policy: advanced accelerated visuomotor policies via matched distillation. arXiv preprint arXiv:2412.09265.
21. Kim et al. (2024) D. Kim, C. Lai, W. Liao, N. Murata, Y. Takida, T. Uesaka, Y. He, Y. Mitsufuji, and S. Ermon Consistency trajectory models: learning probability flow ODE trajectory of diffusion. In International Conference on Learning Representations,
22. Kim et al. (2026) M. J. Kim, Y. Gao, T. Lin, Y. Lin, Y. Ge, G. Lam, P. Liang, S. Song, M. Liu, C. Finn, et al. Cosmos Policy: fine-tuning video models for visuomotor control and planning. arXiv preprint arXiv:2601.16163.
23. Li et al. (2026a) Q. Li, J. Yu, K. Jiang, Y. Wei, Z. Xing, P. Li, R. Chu, S. Zhang, Y. Liu, and Z. Wu DiffusionOPD: a unified perspective of on-policy distillation in diffusion models. In SIGGRAPH Asia 2026 Conference Papers,
24. Li et al. (2026b) S. Li, L. Sun, and Y. Chen One-step flow policy: self-distillation for fast visuomotor policies. arXiv preprint arXiv:2603.12480.
25. Lipman et al. (2023) Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations, External Links: Link
26. Liu et al. (2023) B. Liu, Y. Zhu, C. Gao, Y. Feng, Q. Liu, Y. Zhu, and P. Stone LIBERO: benchmarking knowledge transfer for lifelong robot learning. Advances in Neural Information Processing Systems 36, pp. 44776–44791.
27. Liu et al. (2025) J. Liu, G. Liu, J. Liang, Y. Li, J. Liu, X. Wang, P. Wan, D. Zhang, and W. Ouyang Flow-GRPO: training flow matching models via online RL. In Advances in Neural Information Processing Systems, Vol. 38.
28. Liu et al. (2024) S. Liu, L. Wu, B. Li, H. Tan, H. Chen, Z. Wang, K. Xu, H. Su, and J. Zhu RDT-1B: a diffusion foundation model for bimanual manipulation. arXiv preprint arXiv:2410.07864.
29. Luan et al. (2026) W. Luan, J. Li, W. Zhao, W. Zhang, T. Wu, and R. Ma SnapFlow: one-step action generation for flow-matching VLAs via progressive self-distillation. arXiv preprint arXiv:2604.05656.
30. Mansourian et al. (2025) A. M. Mansourian, R. Ahmadi, M. Ghafouri, A. M. Babaei, E. B. Golezani, Z. Y. Ghamchi, V. Ramezanian, A. Taherian, K. Dinashi, A. Miri, et al. A comprehensive survey on knowledge distillation. arXiv preprint arXiv:2503.12067.
31. Octo Model Team et al. (2024) Octo Model Team, D. Ghosh, H. Walke, K. Pertsch, K. Black, O. Mees, S. Dasari, J. Hejna, C. Xu, J. Luo, T. Kreiman, Y. L. Tan, L. Y. Chen, P. Sanketi, Q. Vuong, T. Xiao, D. Sadigh, C. Finn, and S. Levine Octo: an open-source generalist robot policy. In Proceedings of Robotics: Science and Systems, Delft, Netherlands.
32. Prasad et al. (2024) A. Prasad, K. Lin, J. Wu, L. Zhou, and J. Bohg Consistency policy: accelerated visuomotor policies via consistency distillation. In Robotics: Science and Systems,
33. Reuss et al. (2025) M. Reuss, H. Zhou, M. Rühle, Ö. E. Yağmurlu, F. Otto, and R. Lioutikov FLOWER: democratizing generalist robot policies with efficient vision-language-flow models. In Conference on Robot Learning,
34. Sabour et al. (2025) A. Sabour, S. Fidler, and K. Kreis Align your flow: scaling continuous-time flow map distillation. In Advances in Neural Information Processing Systems, Vol. 38.
35. Sheng et al. (2026) J. Sheng, Z. Wang, P. Li, and M. Liu MP1: MeanFlow tames policy learning in 1-step for robotic manipulation. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 40, pp. 18532–18539.
36. Shukor et al. (2025) M. Shukor, D. Aubakirova, F. Capuano, P. Kooijmans, S. Palma, A. Zouitine, M. Aractingi, C. Pascal, M. Russi, A. Marafioti, S. Alibert, M. Cord, T. Wolf, and R. Cadene SmolVLA: a vision-language-action model for affordable and efficient robotics. arXiv preprint arXiv:2506.01844.
37. Song et al. (2023) Y. Song, P. Dhariwal, M. Chen, and I. Sutskever Consistency models. In International Conference on Machine Learning, pp. 32211–32252.
38. Song et al. (2021) Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, External Links: Link
39. Wang et al. (2025) Z. Wang, Z. Li, A. Mandlekar, Z. Xu, J. Fan, Y. Narang, L. Fan, Y. Zhu, Y. Balaji, M. Zhou, M. Liu, and Y. Zeng One-step diffusion policy: fast visuomotor policies via diffusion distillation. In International Conference on Machine Learning,
40. Wen et al. (2025) J. Wen, Y. Zhu, J. Li, M. Zhu, K. Wu, Z. Xu, R. Cheng, C. Shen, Y. Peng, F. Feng, et al. TinyVLA: towards fast, data-efficient vision-language-action models for robotic manipulation. IEEE Robotics and Automation Letters.
41. Wu et al. (2026) W. Wu, F. Lu, Y. Wang, S. Yang, S. Liu, F. Wang, S. Ma, H. Sun, Y. Wang, Z. Qiu, H. Xiong, Z. Wang, S. Zhou, Y. Ren, K. Zhang, H. Yu, J. Zhao, Q. Zhu, R. Cheng, Y. Li, Y. Huang, X. Zhu, Y. Shen, and K. Zheng A pragmatic vla foundation model. arXiv preprint arXiv:2601.18692v1.
42. Yan et al. (2025) G. Yan, J. Zhu, Y. Deng, S. Yang, R. Qiu, X. Cheng, M. Memmel, R. Krishna, A. Goyal, X. Wang, and D. Fox ManiFlow: a general robot manipulation policy via consistency flow training. In Conference on Robot Learning (CoRL),
43. Ye et al. (2026) S. Ye, Y. Ge, K. Zheng, S. Gao, S. Yu, G. Kurian, S. Indupuru, Y. L. Tan, C. Zhu, J. Xiang, et al. World action models are zero-shot policies. arXiv preprint arXiv:2602.15922.
44. Yin et al. (2024a) T. Yin, M. Gharbi, T. Park, R. Zhang, E. Shechtman, F. Durand, and W. T. Freeman Improved distribution matching distillation for fast image synthesis. Advances in neural information processing systems 37, pp. 47455–47487.
45. Yin et al. (2024b) T. Yin, M. Gharbi, R. Zhang, E. Shechtman, F. Durand, W. T. Freeman, and T. Park One-step diffusion with distribution matching distillation. In 2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 6613–6623.
46. Yuan et al. (2026) T. Yuan, Z. Dong, Y. Liu, and H. Zhao Fast-WAM: do world action models need test-time future imagination?. arXiv preprint arXiv:2603.16666.
47. Zhong et al. (2026) Z. Zhong, H. Yan, J. Li, J. He, T. Zhang, and H. Li VLA-OPD: bridging offline SFT and online RL for vision-language-action models via on-policy distillation. arXiv preprint arXiv:2603.26666.

Appendix A Proofs

See 1

Proof.

For the linear interpolant

{\bm{x}}_{t}=(1-t){\bm{x}}_{0}+t{\bm{x}}_{1},\qquad{\bm{x}}_{0}\sim\mathcal{D},\qquad{\bm{x}}_{1}\sim\mathcal{N}(\mathbf{0},\mathbf{I}),

the score and the corresponding instantaneous velocity field satisfy

\nabla_{{\bm{x}}}\log p_{t}({\bm{x}})=-\frac{1}{t}\left({\bm{x}}+(1-t){\bm{v}}({\bm{x}},t)\right).

Following the flow-to-SDE conversion of Liu et al. (2025), for any diffusion coefficient \sigma_{t}>0, the reverse-time SDE that shares the same marginal p_{t} of PF-ODE is defined as

d{\bm{x}}_{t}=\left({\bm{v}}({\bm{x}}_{t},t)-\frac{\sigma_{t}^{2}}{2}\nabla_{{\bm{x}}}\log p_{t}({\bm{x}}_{t})\right)dt+\sigma_{t}\,d\bar{\mathbf{w}}_{t}.

Substituting Eq. (13) into Eq. (14) gives

d{\bm{x}}_{t}=\left[\left(1+\frac{\sigma_{t}^{2}(1-t)}{2t}\right){\bm{v}}({\bm{x}}_{t},t)+\frac{\sigma_{t}^{2}}{2t}{\bm{x}}_{t}\right]dt+\sigma_{t}\,d\bar{\mathbf{w}}_{t}.

A single Euler–Maruyama step of size \Delta t from \tilde{{\bm{x}}}_{t} therefore defines the Gaussian transition kernel

p_{{\bm{v}}}({\bm{x}}_{t+\Delta t}\mid\tilde{{\bm{x}}}_{t})=\mathcal{N}\left(\mu_{{\bm{v}}},\sigma_{t}^{2}|\Delta t|\,\mathbf{I}_{d}\right),

where

\mu_{{\bm{v}}}=\left(1+\frac{\sigma_{t}^{2}}{2t}\Delta t\right)\tilde{{\bm{x}}}_{t}+\left(1+\frac{\sigma_{t}^{2}(1-t)}{2t}\right){\bm{v}}(\tilde{{\bm{x}}}_{t},t)\Delta t.

For the flow-map parameterization {\bm{f}}_{\theta}({\bm{x}},s,t)={\bm{x}}+(t-s){\bm{u}}_{\theta}({\bm{x}},s,t), its tangent velocity is

\left.\partial_{\tau}{\bm{f}}_{\theta}({\bm{x}},t,\tau)\right|_{\tau=t}={\bm{u}}_{\theta}({\bm{x}},t,t).

We consequently define the student and teacher velocities at \tilde{{\bm{x}}}_{t} by

{\bm{v}}_{S}:={\bm{u}}_{\theta}(\tilde{{\bm{x}}}_{t},t,t),\qquad{\bm{v}}_{T}:={\bm{v}}_{\phi}(\tilde{{\bm{x}}}_{t},t).

Applying Eq. (16) to these two velocities gives

p_{\theta}(\cdot\mid\tilde{{\bm{x}}}_{t})=\mathcal{N}\left(\mu_{S},\sigma_{t}^{2}|\Delta t|\,\mathbf{I}_{d}\right),\qquad p_{\phi}(\cdot\mid\tilde{{\bm{x}}}_{t})=\mathcal{N}\left(\mu_{T},\sigma_{t}^{2}|\Delta t|\,\mathbf{I}_{d}\right).

Their covariances are identical, and the common \tilde{{\bm{x}}}_{t}-dependent term cancels from the difference of their means. Hence,

\mu_{S}-\mu_{T}=\left(1+\frac{\sigma_{t}^{2}(1-t)}{2t}\right)\Delta t\,({\bm{v}}_{S}-{\bm{v}}_{T}).

Using the KL divergence between Gaussians with common covariance, we obtain

\displaystyle D_{\mathrm{KL}}\left(p_{\theta}(\cdot\mid\tilde{{\bm{x}}}_{t})\,\middle\|\,p_{\phi}(\cdot\mid\tilde{{\bm{x}}}_{t})\right) \\ \displaystyle=\frac{\|\mu_{S}-\mu_{T}\|_{2}^{2}}{2\sigma_{t}^{2}|\Delta t|} \\ \displaystyle=\frac{|\Delta t|}{2\sigma_{t}^{2}}\left(1+\frac{\sigma_{t}^{2}(1-t)}{2t}\right)^{2}\|{\bm{v}}_{S}-{\bm{v}}_{T}\|_{2}^{2} \\ \displaystyle=w_{\sigma}(t,\Delta t)\left\|{\bm{u}}_{\theta}(\tilde{{\bm{x}}}_{t},t,t)-{\bm{v}}_{\phi}(\tilde{{\bm{x}}}_{t},t)\right\|_{2}^{2},

where

w_{\sigma}(t,\Delta t):=\frac{|\Delta t|}{2\sigma_{t}^{2}}\left(1+\frac{\sigma_{t}^{2}(1-t)}{2t}\right)^{2}.

∎

See 2

Proof.

Let (s,t,{\bm{x}})\sim q_{\theta} with {\bm{x}}={\bm{x}}_{s}=(1-s){\bm{x}}_{0}+s{\bm{x}}_{1} and m=(s+t)/2. Since s>t, we have |m-s|=(s-t)/2\leq 1/2.

We define the self-consistency target of the student as

T_{\theta}({\bm{x}},s,t)=\frac{1}{2}\big[{\bm{v}}_{\theta}({\bm{x}},s)+{\bm{v}}_{\theta}(\hat{{\bm{x}}}_{m},m)\big],

where \hat{{\bm{x}}}_{m}={\bm{x}}+(m-s)\,{\bm{v}}_{\theta}({\bm{x}},s), and denote the velocity errors at s and m by

e_{s}:=\|{\bm{v}}_{\theta}({\bm{x}},s)-{\bm{v}}_{\phi}({\bm{x}},s)\|_{2},\qquad e_{m}:=\|{\bm{v}}_{\theta}(\hat{{\bm{x}}}_{m},m)-{\bm{v}}_{\phi}(\hat{{\bm{x}}}_{m},m)\|_{2}.

Using the definition of {\bm{u}}_{\phi}, we decompose the difference between the student map and the ideal shortcut as

\displaystyle{\bm{u}}_{\theta}({\bm{x}},s,t)-{\bm{u}}_{\phi}({\bm{x}},s,t) \\ \displaystyle=\big({\bm{u}}_{\theta}({\bm{x}},s,t)-T_{\theta}\big)+\big(T_{\theta}-{\bm{u}}_{\phi}({\bm{x}},s,t)\big) \\ \displaystyle=\big({\bm{u}}_{\theta}({\bm{x}},s,t)-T_{\theta}\big)+\tfrac{1}{2}\big[{\bm{v}}_{\theta}({\bm{x}},s)-{\bm{v}}_{\phi}({\bm{x}},s)\big] \\ \displaystyle\quad+\tfrac{1}{2}\big[{\bm{v}}_{\theta}(\hat{{\bm{x}}}_{m},m)-{\bm{v}}_{\phi}(\hat{{\bm{x}}}_{m},m)\big]+\tfrac{1}{2}\big[{\bm{v}}_{\phi}(\hat{{\bm{x}}}_{m},m)-{\bm{v}}_{\phi}(\bar{\bm{x}}_{m},m)\big],

where the first term is the self-consistency error, the second and third terms are the velocity errors at s and m, respectively, and the last term is the propagated state error induced by the mismatch between the student midpoint \hat{{\bm{x}}}_{m} and the ideal midpoint \bar{{\bm{x}}}_{m}.

By the L-Lipschitzness of the teacher,

\|{\bm{v}}_{\phi}(\hat{{\bm{x}}}_{m},m)-{\bm{v}}_{\phi}(\bar{\bm{x}}_{m},m)\|_{2}\leq L\|\hat{{\bm{x}}}_{m}-\bar{\bm{x}}_{m}\|_{2}=L\,|m-s|\,e_{s}.

Therefore,

\displaystyle\|{\bm{u}}_{\theta}({\bm{x}},s,t)-{\bm{u}}_{\phi}({\bm{x}},s,t)\|_{2}^{2} \\ \displaystyle\leq 4\|{\bm{u}}_{\theta}({\bm{x}},s,t)-T_{\theta}\|_{2}^{2}+e_{s}^{2}+e_{m}^{2}+L^{2}(m-s)^{2}e_{s}^{2} \\ \displaystyle\leq 4\|{\bm{u}}_{\theta}({\bm{x}},s,t)-T_{\theta}\|_{2}^{2}+\Big(1+\frac{L^{2}}{4}\Big)e_{s}^{2}+e_{m}^{2},

where the last inequality uses |m-s|\leq 1/2.

By taking the expectation of Eq. (28) over q_{\theta}, the left-hand side becomes D_{q_{\theta}}({\bm{u}}_{\theta},{\bm{u}}_{\phi}) and the first term on the right-hand side becomes 4\mathcal{L}_{\text{SC}}. The remaining two terms are velocity errors evaluated at (s,{\bm{x}}_{s}) and (m,\hat{{\bm{x}}}_{m}), whereas \mathcal{L}_{\text{OPFD}} averages the same error over (t,\tilde{{\bm{x}}}_{t})\sim\rho_{\theta}.

Let g(t,{\bm{x}}):=\|{\bm{v}}_{\theta}({\bm{x}},t)-{\bm{v}}_{\phi}({\bm{x}},t)\|_{2}^{2}\geq 0, so that \mathcal{L}_{\text{OPFD}}=\mathbb{E}_{\rho_{\theta}}[g].

Then, for any distribution \nu whose density ratio with respect to \rho_{\theta} is at most \kappa,

\mathbb{E}_{\nu}[g]=\mathbb{E}_{\rho_{\theta}}\Big[g\,\frac{d\nu}{d\rho_{\theta}}\Big]\leq\kappa\,\mathbb{E}_{\rho_{\theta}}[g]=\kappa\,\mathcal{L}_{\text{OPFD}}.

Applying this with \nu the distributions of (s,{\bm{x}}_{s}) and (m,\hat{{\bm{x}}}_{m}), which satisfy this condition by the density-ratio assumption, gives \mathbb{E}_{q_{\theta}}[e_{s}^{2}]\leq\kappa\mathcal{L}_{\text{OPFD}} and \mathbb{E}_{q_{\theta}}[e_{m}^{2}]\leq\kappa\mathcal{L}_{\text{OPFD}}.

Therefore,

\displaystyle D_{q_{\theta}}({\bm{u}}_{\theta},{\bm{u}}_{\phi}) \\ \displaystyle\leq 4\,\mathcal{L}_{\text{SC}}+\kappa\Big(1+\frac{L^{2}}{4}\Big)\mathcal{L}_{\text{OPFD}}+\kappa\,\mathcal{L}_{\text{OPFD}} \\ \displaystyle=4\,\mathcal{L}_{\text{SC}}+\kappa\Big(2+\frac{L^{2}}{4}\Big)\mathcal{L}_{\text{OPFD}} \\ \displaystyle\leq\max\Big\{4,\,\frac{\kappa(2+L^{2}/4)}{\lambda}\Big\}\,\mathcal{L}_{\text{FastOPD}} \\ \displaystyle=C_{\lambda}\,\mathcal{L}_{\text{FastOPD}}.

Here, C_{\lambda} grows as \lambda\to 0, reflecting that the self-consistency term alone cannot anchor the student to the teacher.

∎

See 1

Proof.

Let the student and the teacher-induced shortcut model start from the same initial noise sample, {\bm{x}}_{t_{0}}^{\theta}=\bar{\bm{x}}_{t_{0}}\sim\mathcal{N}(\mathbf{0},\mathbf{I}), and follow the same sampling schedule

1=t_{0}>t_{1}>\cdots>t_{K}=0

with step sizes h_{k}:=t_{k}-t_{k+1}>0.

Their rollout states are defined by

{\bm{x}}_{t_{k+1}}^{\theta}={\bm{f}}_{\theta}({\bm{x}}_{t_{k}}^{\theta},t_{k},t_{k+1})\quad\text{and}\quad\bar{\bm{x}}_{t_{k+1}}={\bm{f}}_{\phi}(\bar{\bm{x}}_{t_{k}},t_{k},t_{k+1}),

respectively.

By the definition of {\bm{u}}_{\phi} and m_{k}=(t_{k}+t_{k+1})/2,

{\bm{f}}_{\phi}({\bm{x}},t_{k},t_{k+1})={\bm{x}}-h_{k}{\bm{u}}_{\phi}({\bm{x}},t_{k},t_{k+1})=\bar{\bm{x}}_{m_{k}}-\tfrac{h_{k}}{2}\,{\bm{v}}_{\phi}(\bar{\bm{x}}_{m_{k}},m_{k}),

where \bar{\bm{x}}_{m_{k}}={\bm{x}}-\tfrac{h_{k}}{2}\,{\bm{v}}_{\phi}({\bm{x}},t_{k}).

Thus, {\bm{f}}_{\phi}(\cdot,t_{k},t_{k+1}) is the composition of two Euler steps of size h_{k}/2. Since {\bm{v}}_{\phi}(\cdot,t) is L-Lipschitz, each Euler step is (1+h_{k}L/2)-Lipschitz, and hence {\bm{f}}_{\phi}(\cdot,t_{k},t_{k+1}) is L_{F,k}-Lipschitz with

L_{F,k}:=\Big(1+\frac{h_{k}L}{2}\Big)^{2}.

Then,

\displaystyle\|{\bm{x}}_{t_{k+1}}^{\theta}-\bar{\bm{x}}_{t_{k+1}}\|_{2} \\ \displaystyle=\|{\bm{f}}_{\theta}({\bm{x}}_{t_{k}}^{\theta},t_{k},t_{k+1})-{\bm{f}}_{\phi}(\bar{{\bm{x}}}_{t_{k}},t_{k},t_{k+1})\|_{2} \\ \displaystyle\leq\|{\bm{f}}_{\theta}({\bm{x}}_{t_{k}}^{\theta},t_{k},t_{k+1})-{\bm{f}}_{\phi}({\bm{x}}_{t_{k}}^{\theta},t_{k},t_{k+1})\|_{2} \\ \displaystyle\quad+\|{\bm{f}}_{\phi}({\bm{x}}_{t_{k}}^{\theta},t_{k},t_{k+1})-{\bm{f}}_{\phi}(\bar{\bm{x}}_{t_{k}},t_{k},t_{k+1})\|_{2} \\ \displaystyle\leq h_{k}\|{\bm{u}}_{\theta}({\bm{x}}_{t_{k}}^{\theta},t_{k},t_{k+1})-{\bm{u}}_{\phi}({\bm{x}}_{t_{k}}^{\theta},t_{k},t_{k+1})\|_{2}+L_{F,k}\|{\bm{x}}_{t_{k}}^{\theta}-\bar{\bm{x}}_{t_{k}}\|_{2}.

Denote E_{k}:=\mathbb{E}\big[\|{\bm{x}}_{t_{k}}^{\theta}-\bar{\bm{x}}_{t_{k}}\|_{2}^{2}\big] and D_{k} as in Theorem 1. Then we obtain

E_{k+1}\leq 2h_{k}^{2}D_{k}+2L_{F,k}^{2}E_{k}.

Since the two trajectories start from the same noise sample, E_{0}=0. Unrolling Eq. (36) gives

E_{K}\leq\sum_{k=0}^{K-1}a_{k}D_{k},\qquad a_{k}:=2h_{k}^{2}\prod_{j=k+1}^{K-1}2L_{F,j}^{2}.

By the assumption of \kappa_{\mathrm{roll}} and Proposition 2,

E_{K}\leq\kappa_{\mathrm{roll}}\Big(\sum_{k=0}^{K-1}a_{k}\Big)D_{q_{\theta}}({\bm{u}}_{\theta},{\bm{u}}_{\phi})\leq\kappa_{\mathrm{roll}}\Big(\sum_{k=0}^{K-1}a_{k}\Big)C_{\lambda}\,\mathcal{L}_{\text{FastOPD}}.

The pair ({\bm{x}}_{t_{K}}^{\theta},\bar{\bm{x}}_{t_{K}}) constructed from the same initial noise is a valid coupling of p_{\theta} and p_{\phi}. Therefore,

W_{2}^{2}(p_{\theta},p_{\phi})\leq\mathbb{E}\big[\|{\bm{x}}_{t_{K}}^{\theta}-\bar{\bm{x}}_{t_{K}}\|^{2}_{2}\big]=E_{K}\leq C^{2}\,\mathcal{L}_{\text{FastOPD}},\qquad C^{2}:=\kappa_{\mathrm{roll}}\,C_{\lambda}\sum_{k=0}^{K-1}a_{k}.

Taking square roots on both sides yields

W_{2}(p_{\theta},p_{\phi})\leq C\sqrt{\mathcal{L}_{\text{FastOPD}}},

hence \mathcal{L}_{\text{FastOPD}}\to 0 implies W_{2}(p_{\theta},p_{\phi})\to 0. ∎

Appendix B Experimental Details

B.1 Implementation Details.

All experiments are implemented within the LeRobot framework (Cadene et al., 2024). We report the public checkpoints used for teacher and student initialization in Tab. 7. For two-time projection, the source and target timesteps are embedded using fixed sinusoidal positional encodings, and the remaining weights of the time projection layer \psi(\cdot) are randomly initialized.

B.2 Configuration Details.

In Tab. 8, we report training and evaluation configurations for full reproducibility.

B.3 Computational Resources

For full reproducibility, we report the computational resources used in our experiments in Tab. 9. Latency is measured on a single RTX 3090 (App. B), and real-world policies are deployed on a single RTX 4090 (App. B.5).

B.4 Simulation Experimental Details.

Evaluation uses 50 episodes per task (2,000 episodes on LIBERO; 2,500 per setting on RoboTwin 2.0), and all results are from a single training run (seed 100000). Following the latency details of Luan et al. (2026), we measure end-to-end inference latency on a single NVIDIA RTX 3090 (CUDA 12.8, PyTorch 2.11) in bfloat16 with batch size 1, excluding pre- and post-processing, and report the median of 100 runs after 10 warm-up runs.3

B.5 Real-world Experimental Details

Robot setup. We follow the real-robot setup of MolmoAct2 (Fang et al., 2026a). Specifically, we use the right arm of a bimanual YAM robot with six joints and a parallel-jaw gripper, controlled by absolute joint-position commands (7-dimensional action space), with the joint velocity capped at 0.75 rad/s (Fig. 8). The policy observes two RGB views (top and right wrist; 480\times 640, 30 fps; current frame only) together with the 7-dimensional proprioceptive state (six joints and gripper), and receives the language instruction. Actions and states are normalized with quantile normalization for all models (the gripper is not normalized for MolmoAct2). All policies are run on a single NVIDIA RTX 4090, with SmolVLA-based models in float32.

Dataset. We collect teleoperated demonstrations of multiple tasks on the YAM arm and train all policies on the same dataset (Tab. 10). The evaluation is performed on pnp-plate. The teacher MolmoAct2 is fine-tuned on this dataset for 14,000 iterations. The student is initialized from SmolVLA, warmed up with flow-matching-based behavior cloning for 10,000 iterations, and then trained with FastOPD for 20,000 iterations. At inference, the type of target objects including a ball and a plate are fixed and the language instruction is fixed as “Pick up the ball and place it on the plate”.

Policy execution. Tab. 11 summarizes the execution settings. All policies predict 30-step action chunks (1.0 s) and send commands at 30 Hz. The SmolVLA (warm-up) and our policy share the same synchronous execution scheme. MolmoAct2 is run with its default asynchronous execution, replanning every 0.5 s, skipping the actions already covered by the inference delay, and linearly blending consecutive chunks over 2–10 steps. No real-time chunking or gripper post-processing is applied to any model.

Evaluation details. Each policy is evaluated over 50 episodes with a time limit of 30 s. An episode is counted as successful once the ball has been placed entirely within the plate and the gripper has opened. We report the success rate and the task completion time, the latter averaged over successful episodes only, following the Wang et al. (2025) evaluation protocol. To ensure a fair comparison, the initial scene of every episode is matched to the corresponding episode of a reference evaluation session by aligning the current top-camera frame with the reference image (Fig. 8(c)), so that all policies start from identical object configurations.

Appendix C Additional Experiments

Connection with previous works. Although SnapFlow (Luan et al., 2026) adopts a two-step consistency target similar to ours, the two objectives differ in both their anchor and their scope. SnapFlow is a self-distillation method that combines flow matching on ground-truth actions with a shortcut loss defined only on the fixed interval from t=1 to t=0:

\mathcal{L}_{\text{SnapFlow}}=\lambda_{1}\,\mathcal{L}_{\text{FM}}+(1-\lambda_{1})\lambda_{2}\,\mathcal{L}_{\text{shortcut}},\qquad\mathcal{L}_{\text{shortcut}}=\big\|{\bm{u}}_{\theta}({\bm{x}}_{1},1,0\mid c)-{\bm{v}}_{\text{tgt}}\big\|_{2}^{2},

where {\bm{v}}_{\text{tgt}}=\tfrac{1}{2}\big[\mathrm{sg}[{\bm{v}}_{\theta}({\bm{x}}_{1},1\mid c)]+\mathrm{sg}[{\bm{v}}_{\theta}(\hat{{\bm{x}}}_{0.5},0.5\mid c)]\big] and \hat{{\bm{x}}}_{0.5}={\bm{x}}_{1}-0.5\,\mathrm{sg}[{\bm{v}}_{\theta}({\bm{x}}_{1},1\mid c)]. In contrast, FastOPD enforces self-consistency over arbitrary intervals (s,t), yielding a flow map that supports any number of sampling steps, and anchors the target to the teacher through OPFD rather than to the data through flow matching. SnapFlow can thus be viewed as a teacher-free counterpart of our objective restricted to a single interval, which allows us to isolate the contribution of teacher supervision.

Student initialization. In Sec. 4.3, we studied on-policy distillation (OPD) as an alternative initialization strategy for the flow map. Specifically, we first train the SmolVLA base model with OPD for 5,000 iterations (Stage 1), and then fine-tune it into a flow map with the FastOPD objective for 10,000 iterations (Stage 2). Tab. 12 reports the per-suite results, and Fig. 10 shows the success rate over training iterations. As in Fig. 4, FastOPD consistently improves performance over the course of training under this OPD-based initialization as well.

Appendix D Few-step Distillation algorithms

In this section, we provide detailed descriptions of the few-step distillation algorithms leveraged in our experiments as baselines.

D.1 Distribution Matching Distillation

Referring to OneDP (Wang et al., 2025), we design DMD as a one-step generator, and jointly train it with a fake-score critic {\bm{v}}_{\psi}({\bm{x}},t) of the same architecture, initialized from the same checkpoint as {\bm{v}}_{\theta}. Here, generator and critic are updated alternately at every step.

Generator. A sample is produced with a single Euler step of the student’s own PF-ODE from pure noise {\bm{x}}_{1}\sim\mathcal{N}(\mathbf{0},\mathbf{I}),

\hat{{\bm{x}}}_{0}={\bm{x}}_{1}-{\bm{v}}_{\theta}({\bm{x}}_{1},1).

From the re-noised sample {\bm{x}}_{t} from \hat{{\bm{x}}}_{0}, the teacher and critic denoised estimates are formed as

\hat{{\bm{x}}}_{0}^{\mathrm{real}}={\bm{x}}_{t}-t\,{\bm{v}}_{\phi}({\bm{x}}_{t},t),\qquad\hat{{\bm{x}}}_{0}^{\mathrm{fake}}={\bm{x}}_{t}-t\,{\bm{v}}_{\psi}({\bm{x}}_{t},t).

The distribution-matching direction is then defined as \mathbf{g}_{0}=\hat{{\bm{x}}}_{0}^{\mathrm{fake}}-\hat{{\bm{x}}}_{0}^{\mathrm{real}}. Here, we normalize it per sample by the mean absolute magnitude of the teacher’s own correction,

\mathbf{g}=\frac{\mathbf{g}_{0}}{\mathrm{mean}\big(\big|\hat{{\bm{x}}}_{0}^{\mathrm{real}}-\mathrm{sg}[\hat{{\bm{x}}}_{0}]\big|\big)+\epsilon},

where \mathrm{mean}(\cdot) averages over the action-chunk and action-dimension axes, for training stability. The generator is trained with the surrogate loss whose gradient with respect to \hat{{\bm{x}}}_{0} equals \mathbf{g}:

\mathcal{L}_{\mathrm{gen}}(\theta)=\tfrac{1}{2}\big\|\hat{{\bm{x}}}_{0}-\mathrm{sg}[\hat{{\bm{x}}}_{0}-\mathbf{g}]\big\|_{2}^{2}.

Critic. The fake-score network {\bm{v}}_{\psi} is trained with ordinary denoising score matching on the generator’s output:

\mathcal{L}_{\mathrm{crit}}(\psi)=\mathbb{E}_{t^{\prime},\bm{\epsilon}}\Big[\big\|{\bm{v}}_{\psi}\big({\bm{x}}_{t^{\prime}},t^{\prime}\big)-\big(\bm{\epsilon}-\mathrm{sg}[\hat{{\bm{x}}}_{0}]\big)\big\|_{2}^{2}\Big],

where {\bm{x}}_{t^{\prime}}=(1-t^{\prime})\,\mathrm{sg}[\hat{{\bm{x}}}_{0}]+t^{\prime}\bm{\epsilon}.

D.2 Consistency Trajectory Model

We follow the Consistency Trajectory Model (CTM) formulation of (Kim et al., 2024), which generalizes the original Consistency Distillation (Song et al., 2023) by allowing anytime-to-anytime jump with a denoising score-matching (DSM) regularizer. We train the two-time flow map {\bm{f}}_{\theta}({\bm{x}},s,t)={\bm{x}}+(t-s){\bm{u}}_{\theta}({\bm{x}},s,t) together with an EMA target network \theta^{-}, updated as \theta^{-}\leftarrow\mu\theta^{-}+(1-\mu)\theta.

For sampled timesteps 1\geq s\geq r\geq 0 and an intermediate solver time u\sim\mathrm{Unif}[r,s], with {\bm{x}}_{1}\sim\mathcal{N}(\mathbf{0},\mathbf{I}), {\bm{x}}_{0}\sim p_{0}, and {\bm{x}}_{s}=(1-s){\bm{x}}_{0}+s{\bm{x}}_{1}, the student first jumps from s to r,

\hat{{\bm{x}}}_{r}^{\theta}={\bm{f}}_{\theta}({\bm{x}}_{s},s,r),

while the target integrates the teacher’s PF-ODE from s to u with a 1-step Heun solver, {\bm{x}}_{u}=\mathrm{ODE}_{\phi}({\bm{x}}_{s};s,u), and then estimates from time u to r with the EMA network,

{\bm{x}}_{r}^{-}={\bm{f}}_{\theta^{-}}({\bm{x}}_{u},u,r).

Both are then projected to the clean timestep t=0 through a stop-gradient student G_{\mathrm{sg}}(\cdot,r,0):={\bm{f}}_{\mathrm{sg}[\theta]}(\cdot,r,0), giving \hat{{\bm{x}}}_{0}^{\theta}=G_{\mathrm{sg}}(\hat{{\bm{x}}}_{r}^{\theta},r,0) and {\bm{x}}_{0}^{-}=G_{\mathrm{sg}}({\bm{x}}_{r}^{-},r,0). Thus, the CTM loss is defined as

\mathcal{L}_{\mathrm{CTM}}(\theta)=\mathbb{E}\big\|\hat{{\bm{x}}}_{0}^{\theta}-\mathrm{sg}[{\bm{x}}_{0}^{-}]\big\|_{2}^{2},

and we add a DSM regularizer anchoring the student’s instantaneous velocity {\bm{v}}_{\theta}({\bm{x}}_{s},s) to the teacher’s,4

\mathcal{L}_{\mathrm{DSM}}(\theta)=\mathbb{E}\big\|{\bm{v}}_{\theta}({\bm{x}}_{s},s)-\mathrm{sg}[{\bm{v}}_{\phi}({\bm{x}}_{s},s)]\big\|_{2}^{2},

so the total loss is \mathcal{L}_{\mathrm{}}(\theta)=\mathcal{L}_{\mathrm{CTM}}(\theta)+w_{\mathrm{dsm}}\,\mathcal{L}_{\mathrm{DSM}}(\theta).

D.3 improved MeanFlow

MeanFlow (Geng et al., 2025) parameterizes the average velocity {\bm{u}}({\bm{x}},s,t):=\frac{1}{s-t}\int_{t}^{s}{\bm{v}}({\bm{x}}_{\tau},\tau)\,d\tau and train it through the identity obtained by differentiating the definition along s,

{\bm{u}}({\bm{x}}_{s},s,t)={\bm{v}}({\bm{x}}_{s},s)-(s-t)\,\frac{d}{ds}{\bm{u}}({\bm{x}}_{s},s,t),\qquad\frac{d}{ds}{\bm{u}}=\partial_{\bm{x}}{\bm{u}}\cdot{\bm{v}}({\bm{x}}_{s},s)+\partial_{s}{\bm{u}}.

In the original formulation, {\bm{v}}({\bm{x}}_{s},s) in Eq. (51) is taken to be the student’s own instantaneous velocity {\bm{v}}_{\theta}({\bm{x}}_{s},s):={\bm{u}}_{\theta}({\bm{x}}_{s},s,s), so the regression target is itself computed by the network being optimized. Improved MeanFlow (iMF) (Geng et al., 2026) addresses this issue by reformulating the identity to remove this self-reference.

We follow this reformulation and further adapt it to distillation by replacing the self-referential term with the frozen teacher velocity, {\bm{v}}_{\phi}({\bm{x}}_{s},s), used both as the target and the tangent direction of the total derivative.

Given {\bm{x}}_{1}\sim\mathcal{N}(\mathbf{0},\mathbf{I}), {\bm{x}}_{0}\sim p_{0}, and interpolant {\bm{x}}_{s}, the total derivative is computed by a Jacobian–Vector Product (JVP) of {\bm{u}}_{\theta} at ({\bm{x}}_{s},s,t) along the tangent ({\bm{v}}_{\phi},1,0):

\dot{{\bm{u}}}_{\theta}:=\mathrm{JVP}\big({\bm{u}}_{\theta};\,({\bm{x}}_{s},s,t),\,({\bm{v}}_{\phi},1,0)\big),

The model then predicts one-step velocity as follow:

\hat{{\bm{v}}}_{\theta}={\bm{u}}_{\theta}({\bm{x}}_{s},s,t)+(s-t)\,\mathrm{sg}[\dot{{\bm{u}}}_{\theta}].

Note that MF loss reduces to standard flow matching when t=s. The loss regresses \hat{{\bm{v}}}_{\theta} onto the teacher target with an elementwise Pseudo-Huber loss,

\mathcal{L}_{\mathrm{iMF}}(\theta)=\mathbb{E}\big[\rho_{\delta}(\hat{{\bm{v}}}_{\theta}-{\bm{v}}_{\phi})\big],\qquad\rho_{\delta}(e)=\delta^{2}\Big(\sqrt{1+e^{2}/\delta^{2}}-1\Big),

applied element-wise and averaged over the action dimensions.

Appendix E Detailed Results in Simulation Environments