SplitMeanFlow: Interval Splitting Consistency in Few-Step Generative Modeling

TL;DR

SplitMeanFlow replaces MeanFlow’s differential identity with a purely algebraic one derived from the additivity of the displacement integral: for any split point r ≤ s ≤ t, the average velocity over [r, t] equals the length-weighted sum of average velocities over [r, s] and [s, t]. Enforcing this Interval Splitting Consistency as a self-consistency training objective recovers the MeanFlow Identity in the limit s → t, but eliminates the Jacobian-vector product MeanFlow needs to compute d/dt u. The result is a JVP-free, three-forward-pass training step that the authors deploy in ByteDance’s Doubao TTS — Seed-TTS’s diffusion module — claiming a 20× speedup with one- and two-step students.

Key claims

The additivity of definite integrals implies an exact algebraic identity for the average velocity field: (t-r)·u(z_t, r, t) = (s-r)·u(z_s, r, s) + (t-s)·u(z_t, s, t) for any r ≤ s ≤ t [§4.2.2, Eq. 10].
This algebraic identity strictly generalises the MeanFlow Identity: taking the limit s → t of the rearranged form recovers MeanFlow’s u(z_t, r, t) = v(z_t, t) - (t-r)·d/dt u(z_t, r, t) [§4.3.1, Eqs. 12–18].
The shortcut model is a special case that fixes r = 0 and uses discrete s; SplitMeanFlow’s identity holds for arbitrary continuous-valued r, s, t [§4.3.1, end].
A training iteration is JVP-free — three forward passes (for the three intervals [r,t], [r,s], [s,t]) plus algebraic recombination of the targets, then one standard backward pass [§4.3.2].
The boundary condition u(z_t, t, t) = v(z_t, t) is required to prevent collapse to a trivial solution; in practice this is mixed in via a “flow ratio” α controlling the fraction of batch samples that use the boundary loss [§5.1].
The recipe trains in two stages: a standard Flow Matching teacher first, then a SplitMeanFlow student initialised from the teacher’s weights, with the teacher providing the boundary-condition targets and the student self-supervising the interval-splitting consistency [§5.1].
One- and two-step SplitMeanFlow students are deployed in Doubao TTS (a Seed-TTS-family product), with reported 20× speedup over the multi-step Flow Matching baseline [§1, Abstract].

Method

SplitMeanFlow targets the same bi-time average-velocity field u(z_t, r, t) as MeanFlow, but trains it through an algebraic rather than differential objective. The starting point is the integral definition (t-r)·u(z_t, r, t) = ∫_r^t v(z_τ, τ) dτ together with the additivity property of definite integrals. Substituting “displacement = average velocity × interval length” into the additivity equation yields the Interval Splitting Consistency identity above, which is then divided through by (t-r) to give the per-batch training form u(z_t, r, t) = w_1·u(z_s, r, s) + w_2·u(z_t, s, t) with length-proportional weights.

A training step samples a triple (r, s, t), then runs three forward passes through u_θ — one each for the three intervals — and treats the right-hand-side sum (with stop-gradient) as the regression target for the left-hand side. A separate fraction of batch samples (controlled by flow ratio α) uses the boundary loss u_θ(z_t, t, t) = v_teacher(z_t, t), with the teacher supplying conditional-velocity targets distilled through CFG at a fixed guidance scale. No JVP, no automatic differentiation of intermediate quantities — just three forwards and a standard backward.

Results

The paper’s empirical setting is text-to-speech: SplitMeanFlow is distilled from a Seed-TTS Flow Matching diffusion teacher. The 10-step Flow Matching baseline contributes >50% of total Seed-TTS inference cost, motivating the few-step push. Baselines are Flow Matching (10 NFE + CFG) and DMD; metrics follow Seed-TTS conventions. Headline industrial claim is a 20× speedup for one- and two-step students deployed in Doubao [§1, Abstract]. The paper does not run the standard ImageNet 256² FID benchmark that MeanFlow / SoFlow / AnyFlow compete on, so head-to-head image-generation FID against those methods is not in the paper.

Why it’s interesting

SplitMeanFlow is the third entry in a 2025–2026 sub-thread that asks how to train MeanFlow’s bi-time average-velocity map without the JVP that breaks PyTorch optimisation,, alongside Mean Flows for One-step Generative Modeling (the original differential formulation with JVP) and SoFlow: Solution Flow Models for One-Step Generative Modeling (Taylor-expansion-based consistency, also JVP-free). The three converge on the same parameterisation but disagree on the training objective: MeanFlow uses the differential identity, SoFlow uses a Taylor-expanded local consistency on the solution function, SplitMeanFlow uses the exact algebraic additivity of the integral with no expansion or approximation. The “shortcut model” connection in §4.3.1 makes explicit that prior shortcut-style discrete-step methods are a special case r = 0. For Luma, the relevance is the same as SoFlow: the JVP-free training recipe drops into existing DiT/Flow-Matching stacks, and the production deployment in Doubao TTS is a real signal that the recipe scales to industrial workloads — though the lack of image-FID numbers in the paper means the head-to-head comparison against MeanFlow / SoFlow at matched DiT-XL/2 capacity remains an open question.