High-Order MeanFlow in Generative Modeling

• High-Order MeanFlow is defined by extending traditional mean flow with higher-order derivatives like acceleration to capture nonlinear transition paths.
• The methodology employs both differential and algebraic supervision to ensure interval consistency, resulting in improved sampling stability and model convergence.
• Empirical results demonstrate that incorporating second-order supervision enhances generation quality, computational efficiency, and robustness across various applications.

High-Order MeanFlow denotes the family of techniques and mathematical frameworks extending classical mean flow and flow-matching approaches in generative modeling, filtering, and scientific computation to incorporate higher-order (beyond first-order) dynamical supervision, such as acceleration and path curvature. This enables richer modeling of complex or nonlinear transition paths between probability distributions and more accurate, stable, and efficient solution and inference schemes across a broad range of continuous domains.

1. Mathematical Foundations of High-Order MeanFlow

High-Order MeanFlow builds upon the MeanFlow principle, which replaces modeling of only instantaneous velocity fields $v(z, t)$ with average velocities over intervals: $$u(z_t, r, t) = \frac{1}{t-r} \int_r^t v(z_\tau, \tau) d\tau$$ In its high-order forms, this extends to the average (marginal) acceleration field: $$\bar{a}(z_t, r, t) = \frac{1}{t-r} \int_r^t a(z_\tau, \tau) d\tau$$ with $a(z_t, t)$ the instantaneous acceleration (second derivative), and can generalize to higher time derivatives for arbitrary order.

A key result is the generalized interval consistency condition for average acceleration (Cao et al., 9 Aug 2025): $$(t - r)\bar{a}(z_t, r, t) = (s-r)\bar{a}(z_s, r, s) + (t-s)\bar{a}(z_t, s, t)$$ for any $r < s < t$, analogous to the first-order mean velocity consistency. This ensures that average high-order fields are algebraically and analytically compatible across subintervals, supporting stable one-step or few-step sampling, and providing a principled regression target for model training.

For flow-matching ODEs describing generative transport, high-order expansion is constructed by supervising not only the velocity (first derivative) but also higher derivatives (e.g., acceleration) along the path. In autoregressive transformer settings (HOFAR), this is realized through Taylor expansions of the generative trajectory, with the learning objective including matches to both first- and high-order (e.g., second-order) ground-truth path derivatives (Liang et al., 11 Mar 2025).

2. Algorithmic Realization and Training Strategies

High-order MeanFlow schemes are implemented by extracting both first- and higher (e.g., second) time derivatives at each forward training pass and including their supervision in the loss function. For example, in HOFAR: $$\mathcal{L} = \| \text{First}_{\text{pred}} - \text{First}_{\text{gt}} \| + \| \text{Second}_{\text{pred}} - \text{Second}_{\text{gt}} \|$$ where "First" and "Second" refer to the model's prediction of the velocity and acceleration fields, matched to their analytic targets derived from the trajectory schedule.

In MeanFlow-like models, high-order average fields can be incorporated either differentially (via the chain of derivatives, using Jacobian-Vector Products for practical computation (Liang et al., 11 Mar 2025, Geng et al., 19 May 2025)) or algebraically (using split consistency, which eliminates derivative computation and requires only forward passes (Guo et al., 22 Jul 2025)). The algebraic interval splitting consistency identity is: $$(t - r)u(z_t, r, t) = (s - r)u(z_s, r, s) + (t - s)u(z_t, s, t)$$ with similar forms applying for $\bar{a}$.

Efficient training strategies include curriculum-based objectives as in AlphaFlow, where the interpolation between first-order flow-matching and (possibly) high-order mean consistency is controlled throughout training to alleviate optimization conflicts and improve convergence (Zhang et al., 23 Oct 2025).

3. Statistical Guarantees and Theoretical Optimality

Theoretical analysis demonstrates that high-order mean flow extensions—specifically when learning both velocity and acceleration fields—retain the minimax-optimal statistical convergence rates as first-order flow-matching schemes under standard smoothness assumptions. This is formalized for estimators in Wasserstein and $L_2$ metrics, with convergence rates polynomial in the sample size $N$ determined by the smoothness $s$ of the target density (quantified in Besov spaces) and the data dimension $d$ (Gong et al., 12 Mar 2025): $\text{Estimation error} \leq C\cdot N^{-2s/d}$ Here, adding high-order (acceleration) supervision does not degrade the optimality, and approximation with controlled neural network depth, width, and sparsity ensures that model classes can represent the (high-order) trajectory refinements without incurring statistical inefficiency.

This establishes that second- and higher-order flow-matching/mean flow methods are not just heuristics for faster or higher-quality sampling—they are, under regularity assumptions, as powerful as theoretically possible for distribution estimation (Gong et al., 12 Mar 2025).

4. Expressivity and Computational Complexity

Circuit complexity analysis reveals that high-order MeanFlow generative models, when implemented using Vision Transformers or similar architectures, remain within uniform threshold circuit class $\mathsf{TC}^0$ under reasonable constraints (polylog precision, constant layer count) (Cao et al., 9 Aug 2025). This indicates that increased modeling power is a result of dynamical structure exploitation rather than expressive class expansion.

Key computational advances include the demonstration that approximate attention operations (softmax, kernelized, or low-rank) can be used within high-order MeanFlow models to maintain scalability. For large-scale and high-dimensional settings (e.g., images, video), high-order inference and training can be implemented in $O(n^{2+o(1)})$ time with provably bounded error (Cao et al., 9 Aug 2025). This enables practical application of high-order mean flow modeling without prohibitive cost escalation.

5. Practical Applications and Empirical Impact

Empirical evaluations across multiple domains (image generation, trajectory modeling, filtering, and scientific computing) have demonstrated tangible benefits of high-order MeanFlow methods:

• Generative models (images, video, speech): Incorporation of second-order supervision in both transformer-based and mean flow models consistently improves generation quality (FID, loss curves, perceptual ratings) compared to strictly first-order approaches (Liang et al., 11 Mar 2025, Geng et al., 19 May 2025, Cao et al., 9 Aug 2025).
• Trajectory modeling: High-order supervision enables the modeling of curvature and higher geometric properties of generative paths, crucial for temporal and high-dimensional domains where linear approximations are insufficient (Liang et al., 11 Mar 2025).
• Stability and sample quality: High-order mean flow sampling matches or surpasses multi-step models in efficiency and fidelity, especially when algebraic consistency replaces differential supervision (e.g., SplitMeanFlow) (Guo et al., 22 Jul 2025).
• Efficient filtering: In particle filtering and flow-based updating, high-order expansions yield drift (mean flow) and diffusion terms that fit non-Gaussian, nonlinear posteriors far better than conventional linearized flows (Servadio, 2 May 2025).
• Interpretability and graph-theoretic analysis: High-order flow representations enable the conversion of flow fields into higher-order networks (e.g., FlowHON), supporting memory-aware, physically meaningful graph analysis (Chen et al., 2023).

Methodology	Order	Supervised Quantities	Sampling Steps	Efficiency/Accuracy
Standard Flow Matching	1st	velocity	multi-step	Baseline
HOFAR/High-Order FlowAR	2nd+	velocity + acceleration	multi/few-step	Improved fidelity, low cost
MeanFlow	1st	average velocity	1-step	Efficient, less expressive
High-Order MeanFlow	2nd+	avg. velocity + acc.	1-step/few	Best quality, high efficiency
Particle Flow Filters (DA)	2nd+	drift/diffusion (poly)	analytic	Outperforms linear methods

6. Extensions, Challenges, and Future Directions

Ongoing and future work focuses on:

• Arbitrary-order extensions: Generalized consistency conditions and algebraic regularization principles permit systematic construction of third- and higher-order mean flow models, theoretically enabling arbitrarily accurate path modeling, subject to oracle velocity field complexity and computational tractability (Cao et al., 9 Aug 2025).
• Robustness, stability, and hardware compatibility: Algebraic supervision (removing derivative dependence) ensures generalization and stability across task domains and hardware backends (accelerators lacking higher-order autodiff) (Guo et al., 22 Jul 2025).
• Curriculum and hybrid strategies: Blended objectives (AlphaFlow) mitigate optimization conflicts inherent in joint trajectory flow matching and consistency, accelerating convergence and maximizing performance (Zhang et al., 23 Oct 2025).
• Domain-specific adaptations: For tasks in scientific computing (e.g., CFD, large-eddy simulation) and filtering, high-order mean flows with user-controllable dissipation and stability (e.g., high-order generalized-$\alpha$) enhance robustness and accuracy for physically realistic temporal integration (Deng et al., 2019).
• Graph-theoretic and dynamical representations: Higher-order dependencies in flow-based or particle systems mapped to higher-order network structures offer access to all graph algorithms for flow field analysis (Chen et al., 2023).

Contemporary research thus places high-order MeanFlow approaches at the crossroads of mathematical theory, algorithmic innovation, and practical efficiency, underpinned by proven optimality and broad empirical validation.