Fourier Transformers for Latent Crystallographic Diffusion and Generative Modeling

2026-02-12

AI Summary

This work addresses the challenge of simultaneously enforcing periodic boundary conditions, crystallographic symmetries, and physical constraints in crystal structure generation, as well as the poor scalability of existing methods to large and chemically diverse unit cells. To this end, the authors propose a reciprocal-space-based generative framework that represents species-resolved electron density via truncated Fourier transforms. This representation inherently respects periodicity and space-group symmetries while naturally accommodating variable numbers of atoms. By employing a complex-valued Fourier coefficient Transformer variational autoencoder combined with a latent diffusion model—using only nine basis functions per dimension—the method efficiently reconstructs unit cells containing up to 108 atoms per species on the LeMaterial benchmark and significantly outperforms coordinate-based baselines in unconditional generation tasks for small unit cells (≤16 atoms).

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

The discovery of new crystalline materials calls for generative models that handle periodic boundary conditions, crystallographic symmetries, and physical constraints, while scaling to large and structurally diverse unit cells. We propose a reciprocal-space generative pipeline that represents crystals through a truncated Fourier transform of the species-resolved unit-cell density, rather than modeling atomic coordinates directly. This representation is periodicity-native, admits simple algebraic actions of space-group symmetries, and naturally supports variable atomic multiplicities during generation, addressing a common limitation of particle-based approaches. Using only nine Fourier basis functions per spatial dimension, our approach reconstructs unit cells containing up to 108 atoms per chemical species. We instantiate this pipeline with a transformer variational autoencoder over complex-valued Fourier coefficients, and a latent diffusion model that generates in the compressed latent space. We evaluate reconstruction and latent diffusion on the LeMaterial benchmark and compare unconditional generation against coordinate-based baselines in the small-cell regime ($\leq 16$ atoms per unit cell).

Problem

• crystallographic generative modeling
• periodic boundary conditions
• space-group symmetries
• latent diffusion
• unit-cell representation

Innovation

• Fourier representation
• reciprocal-space generative modeling
• crystallographic symmetry
• latent diffusion
• transformer VAE