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Riemannian Stochastic Interpolants for Amorphous Particle Systems

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

TL;DR

<3-5 sentence high-level summary> The paper tackles the challenge of sampling equilibrium configurations of amorphous materials while preserving correct Boltzmann statistics. It introduces an equivariant Riemannian stochastic interpolant (eRSI) framework that operates on the torus to respect periodic boundary conditions and multi-species symmetries, using an equivariant graph neural network to parametrize the velocity field. The authors prove symmetry preservation for both interpolation and velocity fields, and demonstrate improved generation quality and observable accuracy over baselines, particularly under importance-sampling reweighting. They validate on a 2D metallic glass-former model, showing better fidelity for energy, specific heat, and structure, with scalable behavior as system size grows; limitations include extension to three dimensions and reducing training data requirements.

Abstract

Modern generative models hold great promise for accelerating diverse tasks involving the simulation of physical systems, but they must be adapted to the specific constraints of each domain. Significant progress has been made for biomolecules and crystalline materials. Here, we address amorphous materials (glasses), which are disordered particle systems lacking atomic periodicity. Sampling equilibrium configurations of glass-forming materials is a notoriously slow and difficult task. This obstacle could be overcome by developing a generative framework capable of producing equilibrium configurations with well-defined likelihoods. In this work, we address this challenge by leveraging an equivariant Riemannian stochastic interpolation framework which combines Riemannian stochastic interpolant and equivariant flow matching. Our method rigorously incorporates periodic boundary conditions and the symmetries of multi-component particle systems, adapting an equivariant graph neural network to operate directly on the torus. Our numerical experiments on model amorphous systems demonstrate that enforcing geometric and symmetry constraints significantly improves generative performance.

Riemannian Stochastic Interpolants for Amorphous Particle Systems

TL;DR

<3-5 sentence high-level summary> The paper tackles the challenge of sampling equilibrium configurations of amorphous materials while preserving correct Boltzmann statistics. It introduces an equivariant Riemannian stochastic interpolant (eRSI) framework that operates on the torus to respect periodic boundary conditions and multi-species symmetries, using an equivariant graph neural network to parametrize the velocity field. The authors prove symmetry preservation for both interpolation and velocity fields, and demonstrate improved generation quality and observable accuracy over baselines, particularly under importance-sampling reweighting. They validate on a 2D metallic glass-former model, showing better fidelity for energy, specific heat, and structure, with scalable behavior as system size grows; limitations include extension to three dimensions and reducing training data requirements.

Abstract

Modern generative models hold great promise for accelerating diverse tasks involving the simulation of physical systems, but they must be adapted to the specific constraints of each domain. Significant progress has been made for biomolecules and crystalline materials. Here, we address amorphous materials (glasses), which are disordered particle systems lacking atomic periodicity. Sampling equilibrium configurations of glass-forming materials is a notoriously slow and difficult task. This obstacle could be overcome by developing a generative framework capable of producing equilibrium configurations with well-defined likelihoods. In this work, we address this challenge by leveraging an equivariant Riemannian stochastic interpolation framework which combines Riemannian stochastic interpolant and equivariant flow matching. Our method rigorously incorporates periodic boundary conditions and the symmetries of multi-component particle systems, adapting an equivariant graph neural network to operate directly on the torus. Our numerical experiments on model amorphous systems demonstrate that enforcing geometric and symmetry constraints significantly improves generative performance.

Paper Structure

This paper contains 41 sections, 29 theorems, 112 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Amorphous particle systems
  4. Structure of the configuration space.
  5. Equilibrium distribution.
  6. Invariances on the torus space
  7. Riemannian Stochastic Interpolants
  8. Stochastic interpolants.
  9. Riemannian extensions.
  10. Riemannian Stochastic Interpolants for amorphous materials
  11. Equivariant Riemannian Stochastic Interpolants
  12. Equivariant velocity field on multi-component particle systems
  13. Sampling conditioned on a fixed composition of species.
  14. Fixing composition.
  15. Equivariant optimal transport.
  16. ...and 26 more sections

Key Result

Proposition 1

Given $p_{\mathrm{base}}$ and $p_{\star}$ both $G_{{\mathcal{C}}}$-invariant distributions on ${\mathcal{C}}$ and a $G_{{\mathcal{C}}}$-equivariant interpolant, the interpolation process defined for any $t \in [0,1]$ as $X_t = I(t, X_0, X_1)$ with $X_0 \sim p_{\text{base}}$ and $X_1 \sim p_{\star}$

Figures (13)

  • Figure 1: Illustration of invariance group actions on a configuration of the 2D 10-particle IPL model. The system contains two particle species with different effective diameters (see \ref{['sec:numerics']}). The symmetrized transformation shown corresponds to a $90^{\circ}$ counterclockwise rotation -- equivalently, an axial symmetry with respect to the diagonal from the bottom-left to the top-right corner.
  • Figure 2: Samples from $p_{\star}$ and compared generative models on the 44-particle IPL system. eFM denotes a model trained with standard FM that uses an equivariant velocity respecting system symmetries but ignores the torus geometry, thus generating unphysical particle overlaps near boundaries. RSI uses a non-equivariant velocity field. eRSI is the proposed approach, combining RSI with an equivariant velocity field.
  • Figure 3: Results for $N=10$ particles. (a) Mean potential energy $U$ and (b) specific heat $c_V$ as a function of the number of generated samples $R$ for RSI, eFM, and eRSI. (c–e) Radial distribution function $g(r)$ for the three models, showing target, direct model, and reweighted estimates. RSI fails completely, eFM partially recovers observables, and eRSI performs best.
  • Figure 4: Results for $N=44$ particles. (a) Mean potential energy $U$, and (b) specific heat $c_V$ as a function of the number of generated samples $R$ for RSI, eFM, and eRSI. (c) Radial distribution function $g(r)$ for eRSI averaged over $1.8\times10^6$ samples. RSI and eFM deviate, due to collapsed states (RSI) or boundary overlaps (eFM). RSI samples were of too poor quality to produce $c_V$ estimates. Only eRSI remains consistent with the target distribution.
  • Figure 5: Two trajectories of particle configurations generated by ODE integration with an equivariant velocity field $\hat{v}$ started from symmetric initial configurations (first column). The intermediate configurations are related by the same initial transformation, illustrating \ref{['prop:eq_velocity']}.
  • ...and 8 more figures

Theorems & Definitions (58)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Definition 4
  • Proposition 2
  • Definition 5
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • ...and 48 more