SE(3)-MeanFlow

Generative models of protein backbones — the rigid residue frames that fix a fold before any side chain is placed — are the entry point for de novo protein design. Diffusion and flow-matching methods such as RFdiffusion and Chroma produce designable backbones, but they sample by integrating an ordinary differential equation over hundreds of network evaluations, each requiring a Lie group exponential map on every residue frame. That cost binds hardest on high-throughput design campaigns, whose useful output is thousands of candidate backbones rather than one.

SE(3)-MeanFlow attacks that cost by changing what the network learns. MeanFlow, introduced for Euclidean generative modeling, predicts the average velocity over a time interval rather than the instantaneous velocity, so one evaluation can traverse a long stretch of the trajectory. SE(3)-MeanFlow extends this to the SE(3)^N geometry of protein frames, working natively in the Lie algebra so(3) for rotations and in R^3 for translations. Closed-form average-velocity identities for both components, derived under the decoupled SE(3) = SO(3) × R^3 parametrization, make the training target computable without simulating trajectories and avoid the parallel-transport machinery of earlier Riemannian consistency models.

The work comes from Purdue University with collaborators at Vanderbilt University and Freie Universität Berlin, posted to arXiv in July 2026. It is a preprint awaiting peer review, and no code repository or trained checkpoint has been released; the authors implement the method inside the public QFlow/ReQFlow codebase.

#Key Features

• Simulation-free Lie-group targets: Closed-form average-velocity identities for rotations and translations turn few-step training into direct regression, with no trajectory integration and no parallel transport.
• α-Flow warm-up: The differential MeanFlow target needs a Jacobian-vector product through the backbone network, which is numerically brittle. An SE(3) α-Flow objective removes that product from the rotation branch and serves as a first pretraining stage, after which training switches to a small-time stabilized MeanFlow loss.
• Rectification post-training: A self-reflow stage in the style of ReQFlow yields RecSE3MF, which leads its rectified baseline at every budget tested: 0.968 versus 0.946 designable fraction at 100 steps, 0.894 versus 0.837 at 10.
• Stable geometry under step reduction: Cα validity stays near 0.97 from 100 steps down to 10 and secondary-structure statistics hold steady, so faster sampling does not degrade local backbone geometry.
• Small and cheap to sample: The 16.8M-parameter trunk runs at 43 ms per sampling step on one H100, against 454 ms for the 437M-parameter Riemannian MeanFlow baseline.

#Technical Details

The model keeps the 16.8M-parameter invariant point attention trunk of QFlow/ReQFlow, but swaps the quaternion rotation representation for rotation matrices, conditions on two times via a shared embedding and per-block AdaLN-Zero gating, and predicts trajectory endpoints — changes adding 0.33% to the parameter count. Training uses the SCOPe backbone benchmark of 3,673 preprocessed structures with residue lengths between 60 and 128, the same setup as the baselines it is measured against, for 106k pretraining steps across the two stages plus 6.5k rectification steps on four H100 GPUs.

Evaluation follows the field-standard self-consistency protocol: ten backbones per chain length from 60 to 128, eight sequences designed per backbone with ProteinMPNN, each refolded with ESMFold, and a backbone counted designable when the minimum self-consistency RMSD is at most 2 Å. At 100 steps SE(3)-MeanFlow reaches a designable fraction of 0.936 with mean scRMSD 1.106 Å, against 0.885 for QFlow and 0.832 for Riemannian MeanFlow; at 20 steps it holds 0.867 while QFlow falls to 0.778. The advantage is not universal: at 10 steps the un-rectified model drops to 0.728, below Riemannian MeanFlow's 0.778, and designability comes at a cost in coverage, with pairwise TM-score diversity of 0.415 at 100 steps versus 0.393 for QFlow and 0.343 for Riemannian MeanFlow (lower is better).

#Applications

The target user is anyone generating protein backbones in bulk: de novo design campaigns that filter large candidate pools through downstream sequence design and folding, and screening loops where sampling throughput rather than per-sample quality sets the pace. Cutting several hundred network evaluations to twenty at equal or better designability changes the arithmetic of such a pipeline directly. The objectives are generic to SE(3) frame data and apply to other rigid-body generative problems, though the reported evaluation covers unconditional backbone generation only.

#Impact

The contribution is methodological: few-step consistency training, which reshaped image generation, transfers to the curved geometry of protein frames without approximation, and the stability problems it raises there yield to a JVP-free warm-up. Practical reach is limited for now by scope — a 3,673-structure benchmark subset rather than a large structural corpus, no conditional or motif-scaffolding results, and no public weights — so the work advances a designability-efficiency frontier on a shared benchmark rather than shipping a drop-in tool. The authors name diversity loss as the main open problem and one-step generation as the next target.