Grand Canonical Generators

Andreas Burger; Malte Franke; Luka Mucko; Kjell Jorner; Alan Aspuru-Guzik

cs.LGSep 30, 2026

Grand Canonical Generators

Authors: Andreas Burger, Malte Franke, Luka Mucko, Kjell Jorner, Alan Aspuru-Guzik

Organizations: University of Toronto NVIDIA Vector Institute Toronto, Canada · NVIDIA · ETH Zurich NCCR Catalysis Zurich, Switzerland · University of Toronto Vector Institute Toronto, Canada · University of Toronto NVIDIA Vector Institute Acceleration Consortium CIFAR Toronto, Canada

Abstract

We introduce Grand Canonical Generators (GCG), a generative framework that extends Boltzmann generators to the grand canonical ensemble. We present two designs. The first conditions a variable-size generative model on the chemical potential, sampling particle number and configuration jointly. The second factorizes the grand canonical distribution into a particle-number distribution and the corresponding canonical Boltzmann density. This factorized formulation can use any existing Boltzmann generator for the canonical component, encodes the known linear chemical-potential dependence analytically, and yields a tractable likelihood that supports self-normalized importance sampling (SNIS). Empirically, GCG accurately reproduces grand canonical observables on a Lennard--Jones fluid and methane adsorption in a zeolite, demonstrating generalization across chemical potentials and correction via SNIS and grand canonical Monte Carlo.

Figures & tables

Figure 1: Grand Canonical Generators learn to sample from the $\mu VT$ ensemble, which defines a distribution over both the particle number and geometries, conditioned on the chemical potential $\mu$ .

Figure 2: Lennard–Jones. Curves and bands show the mean and range over three training seeds.

Figure 3: Chemical-potential extrapolation. Mean and $\pm 1$ std over three training seeds.

Figure 4: Inference-time correction of Factorized Grand Canonical Generators on LJ with self-normalized importance sampling (top) and grand canonical Monte Carlo (bottom). Bands show $95\%$ bootstrap intervals (200 resamples for self-normalized importance sampling; $2{,}000$ for grand canonical Monte Carlo over five seeds).

Figure 5: Methane adsorption in MFI zeolite for a Factorized Grand Canonical Generators model, compared to grand canonical Monte Carlo.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: One-coordinate marginals for the periodic ideal gas. Lines are the mean density over the three coordinate axes and bands are their range. Dashed is the analytic uniform density. Morph develops a sharp central peak at $e^{\beta\mu}=1$ . Joint Grand Canonical Generators and Factorized Grand Canonical Generators remain close to the true marginal at all activities.

Figure 7: Ideal-gas errors across activity. Dashed: finite-sample error from independent uniform configurations.

Figure 8: Particle-number transfer to unseen chemical potentials.

Figure 9: Normalized effective sample size $\mathrm{ESS}/M$ of the stratified pool in Fig. 4 . (a) Per fixed- $N$ sector at $400$ configurations, with the largest normalized weight (dashed). (b) Pooled over all sectors for conditional self-normalized importance sampling and grand canonical self-normalized importance sampling as a function of target-energy evaluations.

Figure 10: Inference-time correction with conditional self-normalized importance sampling, grand canonical self-normalized importance sampling, and Bennett acceptance ratio between $N$ and $N+1$ particles. Bennett acceptance ratio combines insertion and deletion energy changes to reconstruct particle-number probabilities.

	Morph	Joint GCG	Factorized GCG	Factorized GCG
System	LJ	LJ	LJ	Zeolite
Temperature	$1.5$	$1.5$	$1.5$	298 K
Box	$6.5^{3}$	$6.5^{3}$	$6.5^{3}$	$40.04{\times}39.80{\times}26.77$ Å
Batch	$1024$	$1024$	$1024$	128
Steps	$60k$	$60k$	$60k$	$200k$
Backbone	$8{\times}512$	$8{\times}512$	$8{\times}512$	$8{\times}768$

Appendix

Table 1: Model hyperparameters for the Lennard–Jones (LJ) fluid and the MFI zeolite.

Explore similar work

Autoregressive Boltzmann Generators
Jun 25, 2026Danyal Rehman, Charlie B. Tan, Yoshua Bengio +2Autoregressive Generation Molecular Generation
Mila – Québec AI Institute · Broad Institute of MIT & Harvard · Aithyra +4
FrOGS: Discrete Neural Sampler for Independent Alloy Configurations Across Chemical Conditions
Sep 1, 2026Kyucheol Min, Elyssa Hofgard, Tess SmidtMarkov Chain Monte Carlo Accelerated Sampling
Massachusetts Institute of Technology Cambridge, MA 02139
Leveraging generative models to assist Monte Carlo sampling
Aug 7, 2026Marylou GabriéGenerative Models Statistical Physics
Laboratoire de Physique Statistique, École normale supérieure, PSL Research University, 24 rue Lhomond, 75005 Paris, France

Jun 25, 2026, cs.LGJ/K move · Enter open · S save

Autoregressive Boltzmann Generators

Danyal Rehman, Charlie B. Tan, Yoshua Bengio, Avishek Joey Bose+1

Mila – Québec AI Institute · Broad Institute of MIT & Harvard · Aithyra+4

Autoregressive Generation Molecular Generation Boltzmann Distributions Zero-Shot Robustness

Open paper AskPDF arXiv

Efficient sampling of molecular systems at thermodynamic equilibrium is a hallmark challenge in statistical physics. This challenge has driven the development of Boltzmann Generators (BGs), which allow rapid generation of uncorrelated equilibrium samples by combining a generative model with exact likelihoods and an importance sampling correction. However, modern BGs predominantly rely on normalizing flows (NFs), which either suffer from limited expressivity due to strict invertibility constraints (discrete time) or computationally expensive likelihoods (continuous time). In this paper, we propose Autoregressive Boltzmann Generators (ArBG) -- a novel autoregressive modelling framework -- that overcomes these limitations by departing from the flow-based BG paradigm. ArBG circumvents the topological constraints of flows and enables sequential inference-time interventions, while offering enhanced scalability by leveraging architectures effective in Large Language Models. We empirically demonstrate that ArBG leads to significant improvements over flow-based models across all benchmarks, but particularly in larger peptide systems such as the 10-residue Chignolin. Furthermore, we introduce Robin, a 132 million parameter transferable model trained with the ArBG framework which improves over the previous state-of-the-art, reducing the zero-shot energy error, E-W

_2

, on 8-residue systems by over 60

\%

. The code can be found at the following link: https://github.com/danyalrehman/autobg.

Abstract

Explore similar work

Autoregressive Boltzmann Generators

FrOGS: Discrete Neural Sampler for Independent Alloy Configurations Across Chemical Conditions

Leveraging generative models to assist Monte Carlo sampling