CrystalGRW: generative modeling of crystal structures with targeted crystallographic properties via geodesic random walks.

Determining whether a candidate crystalline material is thermodynamically stable depends on identifying its true ground-state structure, a central challenge in computational materials science. We introduce CrystalGRW, a diffusion-based generative model on Riemannian manifolds that proposes candidate crystal configurations in stable phases, validated through density functional theory calculations. Our model is designed for de novo generation, which creates crystal structures together with their compositions. The crystal properties, such as fractional coordinates, atomic types, and lattice matrices, are represented on suitable Riemannian manifolds, ensuring that new predictions generated through the diffusion process preserve the periodicity of crystal structures. We also incorporate an equivariant graph neural network to account for rotational and translational symmetries within the model. CrystalGRW generates crystal structures that are stable and closely resemble their density-functional-theory ground states. The model also supports conditional control, such as enforcing a specified crystallographic point group, thereby accelerating materials discovery and inverse design by providing symmetry-consistent, energetically stable candidate crystals for experimental validation.