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Equivariant Diffusion for Crystal Structure Prediction

Peijia Lin, Pin Chen, Rui Jiao, Qing Mo, Jianhuan Cen, Wenbing Huang, Yang Liu, Dan Huang, Yutong Lu

TL;DR

Crystal Structure Prediction (CSP) remains challenging due to complex energy landscapes. The authors introduce EquiCSP, an equivariant diffusion model that enforces lattice permutation and periodic translation invariance while jointly diffusion-lattice parameters and fractional coordinates. They propose Periodic CoM-free Noising and a probabilistic score modeling approach to preserve periodic invariances, along with a dedicated Denoising Model architecture. Empirical results across Perov-5, MP-20, and MPTS-52 datasets show that EquiCSP outperforms prior diffusion-based CSP models and achieves faster convergence. This symmetry-aware framework enhances reliability for ab initio structure generation and materials discovery.

Abstract

In addressing the challenge of Crystal Structure Prediction (CSP), symmetry-aware deep learning models, particularly diffusion models, have been extensively studied, which treat CSP as a conditional generation task. However, ensuring permutation, rotation, and periodic translation equivariance during diffusion process remains incompletely addressed. In this work, we propose EquiCSP, a novel equivariant diffusion-based generative model. We not only address the overlooked issue of lattice permutation equivariance in existing models, but also develop a unique noising algorithm that rigorously maintains periodic translation equivariance throughout both training and inference processes. Our experiments indicate that EquiCSP significantly surpasses existing models in terms of generating accurate structures and demonstrates faster convergence during the training process.

Equivariant Diffusion for Crystal Structure Prediction

TL;DR

Crystal Structure Prediction (CSP) remains challenging due to complex energy landscapes. The authors introduce EquiCSP, an equivariant diffusion model that enforces lattice permutation and periodic translation invariance while jointly diffusion-lattice parameters and fractional coordinates. They propose Periodic CoM-free Noising and a probabilistic score modeling approach to preserve periodic invariances, along with a dedicated Denoising Model architecture. Empirical results across Perov-5, MP-20, and MPTS-52 datasets show that EquiCSP outperforms prior diffusion-based CSP models and achieves faster convergence. This symmetry-aware framework enhances reliability for ab initio structure generation and materials discovery.

Abstract

In addressing the challenge of Crystal Structure Prediction (CSP), symmetry-aware deep learning models, particularly diffusion models, have been extensively studied, which treat CSP as a conditional generation task. However, ensuring permutation, rotation, and periodic translation equivariance during diffusion process remains incompletely addressed. In this work, we propose EquiCSP, a novel equivariant diffusion-based generative model. We not only address the overlooked issue of lattice permutation equivariance in existing models, but also develop a unique noising algorithm that rigorously maintains periodic translation equivariance throughout both training and inference processes. Our experiments indicate that EquiCSP significantly surpasses existing models in terms of generating accurate structures and demonstrates faster convergence during the training process.

Paper Structure

This paper contains 34 sections, 5 theorems, 53 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Methods
  5. Symmetries of Crystal Structure Distribution
  6. An Overview of EquiCSP
  7. Diffusion on Lattice Parameters
  8. Generation
  9. Training
  10. Diffusion on Fractional Coordinates
  11. Generation
  12. Training
  13. The Architecture of the Denoising Model
  14. Experiments
  15. Stable Structure Prediction Results
  16. ...and 19 more sections

Key Result

Proposition 4.5

The marginal distribution $p({\bm{C}}_0)$ by Eq. (eq:ddpm1) is lattice permutation invariant if $\hat{{\bm{\epsilon}}}_{\bm{L}}({\mathcal{M}}_t,t)$ is lattice permutation equivariant, namely $\hat{{\bm{\epsilon}}}_{\bm{L}}({\bm{P}}{\bm{C}}_t,{\bm{P}}{\bm{F}}_t,{\bm{A}},t) ={\bm{P}}\hat{{\bm{\epsilon

Figures (6)

  • Figure 1: (a)$\rightarrow$(b): The lattice permutation of the lattice bases ${\bm{l}}_1,{\bm{l}}_2$. (c)$\rightarrow$(d): The periodic translation of the fractional coordinates ${\bm{f}}_1, {\bm{f}}_2$. (e)$\rightarrow$(f): The schematic diagram of the period translation represented as points on a circle. Both cases do not change the crystal structure. Here, the 2D crystal is used for better illustration.
  • Figure 2: Overview of training process in EquiCSP.
  • Figure 3: The illustration of periodic translation invariance with periodic CoM-free function.
  • Figure 4: Learning curves of lattice loss.
  • Figure 5: Leanring curves of fractional coordinates loss.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Definition 4.1: Composition Permutation Invariance
  • Definition 4.2: O(3) Invariance
  • Definition 4.3: Lattice Permutation Invariance
  • Definition 4.4: Periodic Translation Invariance
  • Proposition 4.5
  • Proposition 4.6
  • Proposition 4.7
  • Definition 1.1
  • Lemma 1.2: xu2021geodiff
  • proof
  • ...and 4 more