Boltzmann generators for amorphous particle systems

Louis Grenioux; Leonardo Galliano; Ludovic Berthier; Giulio Biroli; Marylou Gabrié

stat.MLDec 18, 2025

Boltzmann generators for amorphous particle systems

Authors: Louis Grenioux, Leonardo Galliano, Ludovic Berthier, Giulio Biroli, Marylou Gabrié

Organizations: CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France · Center for Computational Mathematics, Flatiron Institute, New York, NY, USA · Laboratoire de Physique de l’École normale supérieure ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France · Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, 34151, Trieste, Italy ·

Abstract

Sampling configurations in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators address this problem by employing generative models to propose independent configurations, which are then reweighted via importance sampling using exact likelihood evaluations. Recent Boltzmann Generators based on continuous normalizing flows and flow matching have achieved significant success for particle systems and biomolecules. However, these approaches have not been extended to amorphous materials (glasses), for which equilibrium sampling is notoriously slow. Because of their disordered structure, the invariances and geometrical constraints of amorphous materials differ from those of crystals and biomolecules, preventing the direct use of existing generative models. Here, we develop Boltzmann Generators tailored to amorphous materials by building the required equivariances directly into Riemannian stochastic interpolants. Our framework incorporates periodic boundary conditions and particle symmetries using equivariant graph neural networks. Numerical experiments demonstrate that enforcing physical symmetries significantly improves the accuracy of Boltzmann Generators, but also reveal an intrinsic limitation of the continuous-flow formulation: accumulated numerical errors during likelihood integration break time-reversibility, compromising exact thermodynamic reweighting. These results reveal a fundamental challenge for continuous-flow generative models in statistical mechanics and call for alternative approaches that preserve exact thermodynamic consistency.

Figures & tables

Figure 1: Illustration of invariance group actions on a particle configuration. The system contains two particle species with different effective diameters.

Figure 2: Two trajectories of particle configurations generated by integrating Equation 5 with an equivariant velocity field

\hat{v}

started from symmetric initial configurations (first column). The intermediate configurations are related by the same initial transformation.

Figure 3: Physical observables for the

N=10

binary mixture. Radial distribution function

g(r)

computed for (a) RSI, (b) EFM, and (c) ERSI. Dotted, solid colored and solid black lines represent uncorrected model proposals, reweighted estimates, and reference target values, respectively. Convergence of (d) the mean potential energy

\langle U\rangle

and (e) the specific heat

c_{V}

as a function of the number of generated configurations

R

Figure 4: Typical particle configurations for the $N=44$ binary mixture. Snapshots obtained from the target distribution ( $p_{\star}$ ) are compared with proposals generated by the EFM, RSI, and ERSI models. The symmetry-agnostic RSI baseline fails to move particles resulting in random configurations, while the geometry-agnostic EFM model generates realistic local packing but exhibits particle overlaps across the periodic boundaries. The ERSI framework simultaneously preserves both the local liquid structure and the boundary constraints.

Figure 5: Physical observables for the

N=44

binary mixture. Convergence of (a) the mean potential energy

\langle U\rangle

and (b) the specific heat

c_{V}

as a function of the number of generated configurations

R

. (c) Radial distribution function

g(r)

computed from

1.8\times 10^{6}

configurations generated by the ERSI framework. Reference baselines are truncated or omitted where unphysical boundary configurations or numerical instabilities prevent reliable estimate.

Figure 6: Sampling efficiency and energy distribution overlap. (a–b) Normalized effective sample size $\mathrm{ESS}/R$ as a function of the number of generated configurations $R$ for (a) $N=10$ and (b) $N=44$ particles. Results are shown for the symmetry-agnostic RSI baseline (green), the geometry-agnostic EFM baseline (orange), and the proposed ERSI framework (blue). Dashed black lines indicate the asymptotic $1/R$ scaling regime where a single configuration dominates the reweighted estimator. (c–d) Corresponding potential energy histograms $P(U)$ showing the overlap with the equilibrium target reference (dark grey solid curve) for different sample volumes $R$ (indicated by the color bar). For clarity, unphysical high-energy configurations exceeding twice the maximum target energy are omitted from the histograms; this truncation discards $100\%$ , $\approx 84\%$ , and $\approx 3\%$ of the proposals generated at $N=10$ by RSI, EFM, and ERSI, respectively. For $N=44$ , the maximum sample size for RSI and EFM is bounded at $R=8\times 10^{3}$ as generating more samples for these poor baselines does not provide further statistical benefit.

Figure 7: Comparison of model sizes for the ternary mixture at

T=1.0

. (a) Potential energy distribution

P(U)

comparing the target reference (gray) with raw model proposals generated by architectures of varying number of trainable parameters

N_{\theta}

. (b) Radial distribution function

g(r)

comparing the target reference (gray), the unweighted model proposals (dotted lines), and the reweighted estimates (full lines) across different model sizes. (c-d) Convergence of the reweighted mean potential energy

\langle U\rangle

c_{V}

(d) as a function of the number of configurations

R

. (e) Evolution of the normalized effective sample size

\mathrm{ESS}/R

with

R

Figure 8: Sampling efficiency of the ternary mixture in the supercooled regime at

T=0.32

. (a) Potential energy distribution

P(U)

, comparing the target reference (gray) with the raw model proposals for different sample sizes

R

. (b) Radial distribution function

g(r)

comparing the equilibrium reference (gray), the unweighted model proposals (dotted blue), and the reweighted estimates (solid blue). (c–d) Convergence of the reweighted mean potential energy

\langle U\rangle

c_{V}

(d) as a function of the number of generated configurations

R

. (e) Evolution of the normalized effective sample size with

R

settling at approximately

10^{-4}

Figure 9: Energy-filtering protocol. (a) Fraction

\alpha

of lowest-energy samples needed to preserve a given fraction

\mathcal{Z}_{\alpha}/\mathcal{Z}

of the estimated target partition function as a function of the total number of samples

R

. (b) Truncated potential energy histogram showing the discarded high-energy configurations. (c) Distribution of the unnormalized importance weights before and after applying the

10\%

filtering threshold.

Figure 10: Reweighting with filtered samples. (a) Evolution of the normalized effective sample size

\mathrm{ESS}/R

as a function of

R

with energy filtering. (b) Convergence of the reweighted mean potential energy

\langle U\rangle

using the filtered samples. The shaded gray band denotes the equilibrium energy fluctuations, exposing a systematic numerical bias. (c) Convergence of the specific heat

c_{V}

as a function of

R

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

System	$N$	$T$	Model	Architecture $(K\mid\text{HF})$	Batch Size	Training Steps	Learning Rate	Gradient Clipping
Binary Mixture	10	0.10	ERSI	EGNN $3\mid 32$	2048	$5\times 10^{4}$	$1\times 10^{-4}$	2.0
	10	0.10	EFM	EGNN $3\mid 32$	2048	$5\times 10^{4}$	$5\times 10^{-3}$	2.0
	10	0.10	RSI	MLP	2048	$5\times 10^{4}$	$5\times 10^{-4}$	2.0
	44	0.10	ERSI	EGNN $3\mid 32$	1024	$10^{5}$	$5\times 10^{-3}$	2.0
	44	0.10	EFM	EGNN $3\mid 32$	1024	$10^{5}$	$1\times 10^{-3}$	2.0
	44	0.10	RSI	MLP	1024	$10^{5}$	$5\times 10^{-5}$	2.0

Appendix

Table 1: Hyperparameter settings, network architectures

(K\mid\text{HF})

, batch sizes, training steps, learning rates, and gradient clipping thresholds across all studied glass models, state points, and baseline methods.

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Amorphous materials exhibit exceptional mechanical and functional properties, yet their rugged energy landscapes are notoriously difficult to sample. Below the glass-transition temperature, conventional molecular dynamics and Monte Carlo become inefficient because equilibration relies on rare barrier-crossing events, while data-driven generative models are constrained by scarce and biased reference ensembles. Here, we introduce ATLAS, an efficient sampler that learns a diffusion process to generate Boltzmann-distributed amorphous structures directly from a target energy function. Parameterized by an equivariant graph neural network, ATLAS generalizes across system size, temperature, and composition. By exploiting the time reversal of the diffusion process, it enables efficient estimation of thermodynamic quantities and steering toward target observables. In two-dimensional Kob-Andersen systems, ATLAS reproduces parallel tempering Markov chain Monte Carlo structural distributions, free energies and entropies, achieving below 0.2% free energy error in the low-temperature glass regime with over 500-fold fewer energy evaluations. In Cu-Zr and Cr-Co-Ni metallic glasses, ATLAS recovers experimentally observed short-range-order trends and steers structures toward prescribed order parameters and optimized bulk moduli. Moreover, composition-amortized pretraining outperforms composition-specific training from scratch, reduces inverse-design costs by several hundred-fold, and enables sampling with expensive universal machine learning interatomic potentials. Coupled to a large language model agent, ATLAS searches an eight-element space for high-entropy metallic glasses balancing stiffness and ductility, identifying a converged Pareto frontier within 480 oracle evaluations. Together, these results establish ATLAS as a foundation model for sampling, steering and designing amorphous materials.

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