Second-Order MeanFlow: Acceleration-Enhanced Modeling

• Second-Order MeanFlow is an extension of first-order MeanFlow that integrates average acceleration fields to enhance one-step sampling quality and mitigate issues like mode collapse.
• It employs a refined consistency condition by coupling velocity and acceleration targets through derivative-based losses, thereby ensuring dynamical stability and improved distributional coverage.
• The framework is applied broadly in image generation, scientific machine learning, and reinforcement learning, offering efficient, simulation-free, and scalable high-order flow matching.

Second-Order MeanFlow denotes a line of MeanFlow research in which first-order average-velocity transport is extended, refined, or reinterpreted through additional higher-order structure. In the most explicit formulation, it augments MeanFlow with an average acceleration field and a corresponding loss, yielding a theoretically grounded high-order flow-matching model for stable one-step sampling (Cao et al., 9 Aug 2025). In adjacent work, the same label or closely related language is used more broadly for derivative-based consistency objectives, stochastic refinement after a one-step transport, or multi-step compositions of MeanFlow modules rather than literal second derivatives (Huang et al., 31 Jan 2026). The unifying theme is to retain MeanFlow’s simulation-free or low-NFE generation while improving expressivity, distributional coverage, or optimization behavior.

1. MeanFlow background and the motivation for second-order extensions

MeanFlow arises from the observation that one can learn an average velocity over an interval, rather than only an instantaneous velocity field, and thereby perform large updates in generation. In one common formulation, the average velocity over an interval $[r,t]$ is

$$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$

which permits the update

$z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$

MeanFlow therefore jointly concerns instantaneous and interval-level transport, and its objective is designed to learn both fields for few-step or one-step sampling (Kim et al., 24 Nov 2025).

A closely related formulation writes the mean velocity for an interpolation path $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$ as

$$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$

A neural network then approximates this average field through a mean flow matching loss with stop-gradient targets chosen to preserve dynamical consistency (Huang et al., 31 Jan 2026).

The central motivation for second-order variants is a limitation of vanilla one-step MeanFlow. MeanFlow enables efficient, high-fidelity image generation, yet its single-function evaluation generation often cannot yield compelling results; in multimodal settings, 1-NFE MeanFlow may exhibit mode collapse or poor sample quality (Huang et al., 31 Jan 2026). This limitation motivates either a formal high-order extension, in which acceleration is modeled directly, or practical refinements that compensate for the missing variability and consistency of a single deterministic transport.

2. Formal Second-Order MeanFlow: average acceleration and generalized consistency

The paper "Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria" defines Second-Order MeanFlow as an extension that incorporates average acceleration fields into the MeanFlow objective (Cao et al., 9 Aug 2025). For trajectories

$z_t := \alpha_t x + \beta_t \epsilon,$

the marginal velocity and marginal acceleration are

$$v(z_t, t) := \frac{d z_t}{dt}, \qquad a(z_t, t) := \frac{d^2 z_t}{dt^2}.$$

First-order MeanFlow uses the average velocity

$$\bar{v}(z_t, r, t) := \frac{1}{t-r}\int_r^t v(z_\tau,\tau)\, d\tau,$$

whereas Second-Order MeanFlow adds the average acceleration

$$\bar{a}(z_t, r, t) := \frac{1}{t-r}\int_r^t a(z_\tau,\tau)\, d\tau.$$

Its theoretical cornerstone is a generalized consistency condition. A function $f(z_t,r,t)$ is a generalized consistency function if, for all $$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$0,

$$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$1

together with boundary conditions. The paper proves that the average acceleration satisfies the corresponding consistency identity,

$$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$2

with limiting boundary conditions

$$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$3

This establishes the feasibility of learning average acceleration in a way analogous to first-order MeanFlow and is presented as support for stable, one-step sampling and tractable loss functions (Cao et al., 9 Aug 2025).

The loss couples first- and second-order targets: $$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$4 The same work states that these losses are efficiently trainable via Jacobian-vector products and autodiff, and positions the model as a theoretically grounded instance of high-order flow matching (Cao et al., 9 Aug 2025).

3. Complexity-theoretic and algorithmic properties

Second-Order MeanFlow is not presented only as a formal extension; it is also analyzed from the perspective of expressivity and scalable implementation. Under polynomially bounded precision, $$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$5 transformer layers, and $$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$6 sampling steps, the sampling process of both MeanFlow and Second-Order MeanFlow can be computed by uniform threshold circuits of depth $$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$7 and size $$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$8, placing them in the class $$u(z_t, r, t) \triangleq \frac{1}{t-r} \int_r^t v_\tau(z_\tau, \tau)\, d\tau,$$9 (Cao et al., 9 Aug 2025). In the same analysis, multi-head attention with softmax, MLPs, and LayerNorm are treated as operations that can be realized or efficiently approximated within the same complexity regime.

The same paper further addresses the dominant transformer bottleneck, namely attention. It cites fast, provable attention approximation algorithms that achieve entrywise error bounded by $z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$0, and derives a full-model inference guarantee for Second-Order MeanFlow with total runtime

$z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$1

The main efficiency theorem is stated for second-order MeanFlow models with moderate feature dimension $z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$2 and weight bound $z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$3, showing that the model output can be computed with error at most $z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$4 in time $z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$5 (Cao et al., 9 Aug 2025).

These results do not by themselves establish empirical superiority, but they provide a formal account of why high-order MeanFlow can, in principle, combine richer dynamics with practical sampling efficiency. A plausible implication is that the high-order extension is intended to increase dynamical expressivity without abandoning the constant-depth, highly parallel structure that makes MeanFlow attractive for low-NFE generation.

4. Multiple meanings of “second-order” in the MeanFlow literature

Across later MeanFlow papers, “second-order” is not used uniformly. Some works use the phrase for explicit acceleration modeling; others use it for a corrective stochastic increment, a derivative-based consistency term, or a two-step MeanFlow composition. The distinctions are consequential because these constructions are not mathematically equivalent (Huang et al., 31 Jan 2026, Zhang et al., 23 Oct 2025, Sun et al., 16 Nov 2025, Yang et al., 8 Apr 2026).

Usage of “second-order”	Mechanism	Status
Formal high-order MeanFlow	Learns average acceleration in addition to average velocity	Mathematically explicit
Stochastic correction	Adds a noise-injection refinement after 1-NFE transport	Practical refinement
Objective-level second order	Uses derivative-based consistency or refined regression targets	Training/objective interpretation
Multi-step analog	Stacks MeanFlow mappings over sub-intervals	Not a true second derivative

In RMFlow, the “second” component is a noise-injection refinement after a coarse 1-NFE MeanFlow transport. The practical one-step formula is

$z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$6

and the full objective is

$z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$7

The paper explicitly interprets the second noise step as a correction to deterministic mean flow transport, “reminiscent of ‘second-order’ schemes in numerical ODEs,” but it is a stochastic refinement rather than an acceleration-field model (Huang et al., 31 Jan 2026).

AlphaFlow uses second-order language at the level of the MeanFlow objective itself. It shows that the MeanFlow loss decomposes into trajectory flow matching and trajectory consistency, with the derivative term acting as a consistency component that makes the loss more expressive but also harder to optimize (Zhang et al., 23 Oct 2025). MENO similarly characterizes its improved MeanFlow objective as a refined or second-order method because the target depends on $z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$8 and $z_r = z_t - (t-r)\,u_\theta(z_t, r, t).$9, while still performing one-step decoding at inference (Yang et al., 8 Apr 2026).

By contrast, MFI-ResNet explicitly states that it does not employ second derivatives or acceleration fields. Its “higher-order” behavior is realized by stacking two MeanFlow modules in stage 4, performing a two-step continuous transformation; the paper states that this is “not a true ‘second-order’ ODE in the mathematical sense” (Sun et al., 16 Nov 2025). The literature therefore supports a narrow definition of Second-Order MeanFlow based on average acceleration and a broader, looser usage centered on corrective or multi-stage MeanFlow mechanisms.

5. Optimization, curriculum design, and training stability

A major theme in the MeanFlow literature is that interval-level or derivative-enriched objectives are difficult to optimize. "Understanding, Accelerating, and Improving MeanFlow Training" analyzes the interaction between instantaneous velocity $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$0 and average velocity $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$1, reporting three findings: well-established instantaneous velocity is a prerequisite for learning average velocity; learning of instantaneous velocity benefits from average velocity when the temporal gap is small, but degrades as the gap increases; and smooth learning of large-gap average velocities depends on the prior formation of accurate instantaneous and small-gap average velocities (Kim et al., 24 Nov 2025). On that basis, it proposes acceleration of $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$2-learning using methods such as MinSNR and DTD, followed by a progressive weighting of $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$3-supervision from small to large temporal gaps, with

$\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$4

Empirically, with the same DiT-XL backbone, this training scheme reaches FID $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$5 on 1-NFE ImageNet $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$6, compared to $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$7 for the conventional MeanFlow baseline, and matches the baseline with $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$8 shorter training time (Kim et al., 24 Nov 2025).

AlphaFlow studies the same instability from a different angle. It decomposes the MeanFlow objective into trajectory flow matching and trajectory consistency and reports that their gradients are strongly negatively correlated, with cosine similarity typically below $\bm{x}_t = a_t \bm{x}_1 + b_t \bm{x}_0$9 during training (Zhang et al., 23 Oct 2025). Its remedy is an $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$0-Flow curriculum that anneals from trajectory flow matching toward MeanFlow: $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$1 This curriculum reduces optimization conflict and improves convergence; on class-conditional ImageNet-1K $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$2 with vanilla DiT backbones, the largest model achieves FID $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$3 at 1-NFE and $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$4 at 2-NFE (Zhang et al., 23 Oct 2025).

A third line of work replaces the continuous MeanFlow identity with a discretized curriculum. "Discrete Meanflow Training Curriculum" derives a discrete recursion

$$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$5

and uses it to stage training from easy, coarse objectives to the full MeanFlow target only at the end (Hsu et al., 10 Apr 2026). Initialized with a pretrained Flow Model, the curriculum reaches one-step FID $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$6 on CIFAR-10 in only $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$7 epochs; the same work reports per-batch speedups of about $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$8 on CIFAR-10 and about $$\bm{u}_{t,r}(\bm{x}_t) := \frac{\bm{x}_r-\bm{x}_t}{r-t} = \frac{1}{r-t}\int_t^r \bm{u}_s(\bm{x}_s)\, ds.$$9 on ImageNet $z_t := \alpha_t x + \beta_t \epsilon,$0 latents relative to full MeanFlow training (Hsu et al., 10 Apr 2026).

Taken together, these studies show that “second-order” behavior in MeanFlow is as much an optimization problem as a modeling problem. Large-gap average transport, trajectory consistency, and derivative-based targets are learnable, but they typically require curriculum structure, conflict mitigation, or discrete approximations.

6. Applications, empirical scope, and related extensions

Practical variants associated with second-order MeanFlow ideas have been evaluated across multimodal generation, scientific machine learning, manifold-valued generation, and reinforcement learning. RMFlow is the clearest multimodal example: it integrates a coarse 1-NFE MeanFlow transport with a tailored noise-injection refinement step and achieves near state-of-the-art results on text-to-image, context-to-molecule, and time-series generation using only 1-NFE at computational cost comparable to baseline MeanFlows (Huang et al., 31 Jan 2026). On synthetic density tasks it achieves lower TV and KL distance to ground truth than 1-NFE MeanFlow and nearly matches 8-NFE and 32-NFE MeanFlow; on QM9, 1-NFE RMFlow greatly improves validity and stability over 1-NFE MeanFlow, with molecule stability metrics on par with state-of-the-art multi-step diffusion methods; and on Lorenz, FitzHugh-Nagumo, and COCO it significantly outperforms single-step MeanFlow while narrowing the gap to multi-step and diffusion models (Huang et al., 31 Jan 2026).

Domain-specific generalizations preserve the same ambition—efficient, high-quality transport—but adapt the MeanFlow identity to non-Euclidean or heterogeneous settings. Riemannian MeanFlow extends MeanFlow to smooth manifolds by defining average velocity through parallel transport and derives an intrinsic identity linking average and instantaneous velocities; it reports competitive one-step sampling with improved quality-efficiency trade-offs and substantially reduced sampling cost on spheres, tori, and $z_t := \alpha_t x + \beta_t \epsilon,$1 (Zhong et al., 11 Mar 2026). Equivariant MeanFlow for molecular graphs introduces synchronized MeanFlow dynamics for discrete topology and continuous geometry with a unified time bridge, mutual conditioning, and an SE(3)-equivariant design, and reports consistent gains in generation quality, physical validity, and sampling efficiency (Xu et al., 9 Apr 2026).

Other applications emphasize that MeanFlow-style one-step mappings can serve as efficient correctors or policies even when the formal acceleration-field interpretation is absent. MENO uses an improved MeanFlow decoder to restore small-scale details and large-scale dynamics in neural operators, improving power spectrum density accuracy by up to a factor of $z_t := \alpha_t x + \beta_t \epsilon,$2 and achieving $z_t := \alpha_t x + \beta_t \epsilon,$3 faster inference than DDIM-enhanced counterparts (Yang et al., 8 Apr 2026). In online control, DMPO combines MeanFlow, dispersive regularization, and RL fine-tuning, reporting $z_t := \alpha_t x + \beta_t \epsilon,$4Hz control and $z_t := \alpha_t x + \beta_t \epsilon,$5-$z_t := \alpha_t x + \beta_t \epsilon,$6 inference speedup (Zou et al., 28 Jan 2026), while SOM constructs MeanFlow targets from the Q-function through score estimation and a probability flow ODE and achieves state-of-the-art performance on locomotion tasks with a single generation step (Kim et al., 22 May 2026).

This range of results suggests that the importance of Second-Order MeanFlow lies less in a single fixed architecture than in a family of techniques for enriching one-step transport. In its strict sense, it is the acceleration-augmented extension defined in (Cao et al., 9 Aug 2025). In broader usage, it names a set of corrective, derivative-aware, or multi-stage MeanFlow mechanisms that seek to recover multimodality, stabilize optimization, or preserve efficiency when one-step generation is pushed beyond the capabilities of first-order average transport alone.