Fourier Transformers for Latent Crystallographic Diffusion and Generative Modeling

TL;DR

This paper introduces a Fourier-based generative model for crystalline materials that naturally incorporates symmetry and periodicity, enabling efficient generation of large and diverse crystal structures.

Contribution

It presents a reciprocal-space generative pipeline using Fourier transforms and a transformer VAE with latent diffusion, addressing limitations of coordinate-based methods.

Findings

Reconstructs unit cells with up to 108 atoms per species using only nine Fourier basis functions per dimension.

Demonstrates effective unconditional generation in the small-cell regime compared to coordinate-based baselines.

Evaluates reconstruction and generation quality on the LeMaterial benchmark.

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).