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ESS-Flow: training-free guidance as Bayesian inference in source space

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling

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Proposes ESS-Flow, a gradient-free, training-free controlled generation method that runs elliptical slice sampling in the Gaussian source space of pretrained flow-based models, evaluated on materials design and protein structure prediction from sparse distances.

Problem

Controlled generation and Bayesian inverse problems with pretrained flow-based models often need gradients through the transport map and potential, which are unreliable or unavailable under quantization or non-differentiable simulators.

Approach

Recast the target posterior as a pullback potential g(T(z)) times the Gaussian source prior, so sampling needs no Jacobians; elliptical slice sampling then explores the source space using only point-wise forward evaluations of the transport map and potential.

Key results

• On material generation, ESS-Flow reduces bulk modulus MAE to 9.05 GPa vs 26.67 for DAPS and 50.98 for PnP-Flow, and band gap MAE to 0.50 eV vs 3.34 for DAPS.
• On protein structure prediction from 300 sparse distances, ESS-Flow reaches ELBO 8.89, versus -8.07 for DAPS and -5.68 for ADP-3D, indicating more realistic structures, though its L2 distance d_y (37.02) is much worse than ADP-3D (3.43).
• Multi-fidelity importance re-weighting gives effective sample size of 68.1% and 74.1% for bulk and shear modulus but only 6.8% and 11.7% for band gap and stability.

Surprising findings

• ADP-3D and DAPS achieve much lower observation distance d_y but produce unrealistic, collapsed protein structures according to ELBO, showing data-fit and realism trade-off.
• ESS-Flow's S.U.N. rates are low (6.1% to 21.6%), reflecting extreme targets set at the 99th percentile.

Limitations

• Performs poorly when the prior poorly covers the target, e.g. noiseless image inpainting or lower-dimensional target manifolds.
• Tends to get trapped in modes on protein task, requiring parallel chains.
• Importance-reweighted multi-fidelity samples have low effective sample size for sharp targets.

Experimental setup

Datasets: MP-20 JARVIS-DFT PDB:7r5b

Benchmarks: MP-20 JARVIS-DFT PDB:7r5b

Metrics: absolute error to target property energy above hull validity stability uniqueness novelty S.U.N. rate L2 distance to observed distances RMSD to ground truth ELBO

Baselines: D-Flow PnP-Flow ADP-3D DAPS Unconditional

Builds on: elliptical slice sampling D-Flow PnP-Flow DAPS FlowMM

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• AlphaFold Meets Flow Matching for Generating Protein Ensembles (ICML 2024 · similarity 0.88)
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