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RESEARCH ·NeurIPS 2026

SSD: Shell-Guided Spherical Diffusion

以「球殼先驗」取代高斯雜訊的分子 3D 構型生成

Molecular diffusion models start from a Gaussian cloud whose size is set by dimension, not chemistry. SSD starts every molecule on a chemically scaled spherical shell and guides it inward with shell-aware drifts — a plug-in that upgrades five coordinate-space backbones without touching their architectures, losses or training budgets.

NeurIPS 2026 · accepted interactive — scroll to step through the method
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Background

Why the starting point of a molecular diffusion model matters.

Generating a molecule's 3D geometry — where every atom sits in space — is a core step in drug discovery and computational chemistry. Modern generators (GeoDiff, EDM, SubGDiff, flow-matching models…) do it the diffusion way: start from random noise and denoise it, step by step, into a valid structure.

Almost all of them draw that starting noise from an isotropic Gaussian. In high dimensions a Gaussian isn't a blob — it concentrates on a thin shell of radius σT√3n (the Gaussian Annulus Theorem). That radius depends only on the noise level and the atom count n, not on how big molecules actually are. The bigger the molecule, the worse the mismatch — and the longer, more wandering the denoising trajectory.

SSD keeps the shell, but makes it the right size: a dataset-calibrated radius, plus forward and reverse dynamics designed around that shell. Neither piece alone is enough; together they give short, stable shell-to-structure paths.

▸ key terms

Prior — the distribution a generator starts sampling from. Drift — the deterministic push applied to each atom at every step. SE(3)-equivariance — rotating or moving the input rotates/moves the output identically; molecules have no preferred orientation. COV / MAT — coverage (diversity, ↑) and matching RMSD (quality, ↓) of generated conformers against references.

GEOMETRY · shell → structure
Gaussian radius r ≈ σT√3n set by dimension molecule's size rchem ≪ σT√3n set by chemistry scale mismatch → long, wandering denoising paths rSSD dataset-calibrated radius QM9 · 6.37 Å Drugs · 21.16 Å x = r · v / ‖v‖, v ~ N(0, I) re-centred · random π every atom starts on a shell the size of real molecules distance to target vs. t SSD · linear, same arrival OU · exponential, spread f = αt · (xTπ(i) − xt) / ‖xTπ(i) − xt‖ normalised pull → uniform speed, thin forward shell vrad inward pull vrep d < dmin vscore SE(3)-eq. dx = (vrad + vrep + vscore) dt + σt dw̄ closed-form drifts · no new learnable parameters backbone + SSD GeoDiffDrugs · COV-R 64.1 91.7 SubGDiffDrugs · COV-R 76.2 92.5 EDMQM9 · Mol-Stab 82.0 99.8 SemlaFlowDrugs · Validity 93.9 100 MCFDrugs · COV-P 66.8 68.8 same architecture · same loss · same training & sampling budget weaker backbones (GeoDiff, EDM) catch up with far stronger ones 246.9 → 1.71 path-excess ratio · SubGDiff → SSD-SubGDiff (QM9) +32.5 COV-R on ≥50-atom GEOM-Drugs molecules 5 backbones · 0 new params faster convergence · robust at 20 steps gains grow with molecule size — where Gaussian priors fail most
A Gaussian prior concentrates at radius σT√3n — set by dimension, far larger than the molecule itself.
STEP 00 · the problem

The Gaussian prior is the wrong size

Coordinate-space diffusion models start from N(0, σT²I). In 3n dimensions that noise sits on a thin shell of radius σT√3n — fixed by the noise level and atom count, not by chemistry.

The sampler must then drift a long, wandering way back to a molecule-sized structure — and the mismatch grows with n.

STEP 01 · shell prior

Start on a chemically scaled shell

SSD places every atom on a spherical shell of radius rSSD, calibrated to the mean radius of centred training conformations — 6.37 Å on QM9, 21.16 Å on Drugs.

It is one dataset-level constant (a per-molecule radius would break the prior's role as a marginal). Atoms get a random shell assignment π per trajectory — deterministic matching collapses coverage.

STEP 02 · forward process

A uniform-speed pull onto the shell

The forward SDE replaces Gaussian drift with a normalised directional attraction: each atom moves toward its assigned shell point at speed αt, whatever its distance.

Unlike Ornstein–Uhlenbeck's exponential pull, every atom arrives on the same schedule — a thin forward shell, so training noise already looks like SSD's prior.

STEP 03 · reverse process

Three structured drifts, back to a molecule

Sampling starts from the same shell and combines radial attraction (a uniform inward pull), short-range repulsion (atoms never closer than dmin) and an SE(3)-equivariant score that refines local geometry.

The added drifts are closed-form and depend only on directions and pairwise offsets, so equivariance is preserved and no learnable parameters are added.

STEP 04 · plug-in

One framework, five backbones

SSD drops into GeoDiff, SubGDiff, EDM, SemlaFlow-style flow matching and MCF — each evaluated under its own canonical protocol, with identical training and sampling budgets.

Every backbone improves; GeoDiff on Drugs jumps from 64.1 → 91.7 COV-R, and EDM's QM9 molecule stability from 82.0 → 99.8%. The ablation confirms neither the shell nor the drifts alone get there.

STEP 05 · the result

Shorter paths, bigger molecules

Trajectories become direct: SubGDiff's path-excess ratio drops from 246.9 to 1.71. SSD converges faster in training, stays strong at just 20 sampling steps, and its gains grow with size — +32.5 COV-R on ≥50-atom drug-like molecules.