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Scalable Equilibrium Sampling with Sequential Boltzmann Generators
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Introduces Sequential Boltzmann Generators (SBG), pairing a non-equivariant TarFlow normalizing flow (softly SE(3)-equivariant via augmentation) with annealed Langevin transport and AIS/SMC reweighting to sample peptides up to hexapeptide (and chignolin) in Cartesian coordinates.
Problem
Equilibrium sampling of molecular Boltzmann distributions in Cartesian coordinates is intractable for existing Boltzmann generators beyond dipeptides, because equivariant CNF-based proposals have poor target overlap and low importance-sampling ESS.
Approach
Replace equivariant continuous flows with a fast exactly invertible non-equivariant transformer flow (TarFlow) trained with soft SE(3) equivariance via augmentation and a center-of-mass noise correction, then improve samples at inference by annealed Langevin transport from the flow energy to the target energy with continuous-time AIS/SMC importance weights.
Key results
- On alanine dipeptide, ECNF++ achieves the best T-W2 (0.189) and ESS (0.275) among methods, while SBG AIS achieves the best E-W2 among SBG variants at 0.630 vs ECNF's 0.419 (ECNF best E-W2).
- On trialanine, SBG AIS reaches ESS 0.052 vs 0.003 for ECNF++ and T-W2 0.450 vs 0.962.
- On hexa-alanine, SBG AIS reduces E-W2 to 1.021 vs 10.668 for ECNF++ and T-W2 to 1.431 vs 1.902; SBG is the first to scale to chignolin (decapeptide) where SMC E-W2 is 3.571 vs SNIS 12.046.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SBG AIS instead of ECNF++ with SNIS | Alanine dipeptide sampling · Alanine dipeptide | \(\mathcal{E}\text{‑}\mathcal{W}_2\): 0.914 $\pm$ 0.122 -> 0.630 $\pm$ 0.249 | Helped |
| Use SBG AIS instead of ECNF++ with SNIS | Alanine dipeptide sampling · Alanine dipeptide | ESS: 0.275 $\pm$ 0.010 -> 0.030 $\pm$ 0.012 | Hurt |
| Use SBG AIS instead of ECNF++ with SNIS | Alanine dipeptide sampling · Alanine dipeptide | \(\mathbb{T}\text{‑}\mathcal{W}_2\): 0.189 $\pm$ 0.019 -> 0.418 $\pm$ 0.090 | Hurt |
| Use SBG AIS instead of ECNF++ with SNIS | Trialanine sampling · Trialanine | ESS: 0.003 $\pm$ 0.002 -> 0.052 $\pm$ 0.013 | Helped |
| Use SBG AIS instead of ECNF++ with SNIS | Trialanine sampling · Trialanine | \(\mathcal{E}\text{‑}\mathcal{W}_2\): 2.206 $\pm$ 0.813 -> 0.797 $\pm$ 0.094 | Helped |
| Use SBG AIS instead of ECNF++ with SNIS | Trialanine sampling · Trialanine | \(\mathbb{T}\text{‑}\mathcal{W}_2\): 0.962 $\pm$ 0.253 -> 0.450 $\pm$ 0.043 | Helped |
| Use SBG AIS instead of ECNF++ with SNIS | Alanine tetrapeptide sampling · Alanine tetrapeptide | ESS: 0.016 $\pm$ 0.001 -> 0.046 $\pm$ 0.014 | Helped |
| Use SBG AIS instead of ECNF++ with SNIS | Alanine tetrapeptide sampling · Alanine tetrapeptide | \(\mathcal{E}\text{‑}\mathcal{W}_2\): 5.638 $\pm$ 0.483 -> 0.883 $\pm$ 0.213 | Helped |
| Use SBG AIS instead of ECNF++ with SNIS | Hexa-alanine sampling · Hexa-alanine | \(\mathcal{E}\text{‑}\mathcal{W}_2\): 10.668 $\pm$ 0.285 -> 1.021 $\pm$ 0.239 | Helped |
| Use SBG AIS instead of ECNF++ with SNIS | Hexa-alanine sampling · Hexa-alanine | ESS: 0.006 $\pm$ 0.001 -> 0.034 $\pm$ 0.015 | Helped |
| Use SBG SMC instead of SNIS | Chignolin sampling · Chignolin | \(\mathcal{E}\text{‑}\mathcal{W}_2\): 12.046 -> 3.571 | Helped |
| Add center-of-mass adjusted proposal energy (ablation) | SBG SMC · Trialanine | \(\mathbb{T}\text{‑}\mathcal{W}_2\) | Helped |
Surprising findings
- On alanine dipeptide the ECNF++ baseline outperforms both SBG variants on ESS and T-W2, contrary to the overall trend.
- SE(3)-EACF's SNIS performance degraded sharply when retrained at 300K rather than the original 800K.
Limitations
- Continuous normalizing flows (flow matching) cannot easily be used with the annealing/AIS scheme because of expensive ODE simulation and divergence computation, limiting modern flow-matching methods.
- Only alanine-based peptides up to 6 residues plus chignolin are evaluated; decapeptide results are reported only qualitatively for SBG SMC with no baselines.
- Several baselines (SE(3)-EACF, ECNF) fail or are unscalable beyond dipeptides, and the ESS of SBG remains low (0.03-0.05) on several systems.
Experimental setup
Datasets: alanine dipeptide MD (300K) trialanine MD alanine tetrapeptide MD hexa-alanine MD chignolin MD
Benchmarks: alanine dipeptide trialanine alanine tetrapeptide hexa-alanine chignolin
Metrics: ESS energy W2 (E-W2) torus dihedral W2 (T-W2) TICA W2
Baselines: SE(3)-EACF ECNF ECNF++ SNIS reweighting
Builds on: Boltzmann generators TarFlow annealed importance sampling sequential Monte Carlo transferable Boltzmann generators
Closest papers by meaning
- Temperature-Annealed Boltzmann Generators (ICML 2025 · similarity 0.91)
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