Crystal generative models typically use variational auto-encoders (VAEs) , generative adversarial networks (GANs) , genetic algorithms , or substitution . One example of a crystal generative model based on VAE is the Fourier-Transformed Crystal Properties (FTCP) representation . The FTCP representation is a crystal representation that not only incorporated both chemical and structural crystal properties but was invertible as well (property → chemistry + structure). When combined with a jointly trained VAE model, a probabilistic property-structured latent space was obtained that allowed for the generation of novel crystals with user-defined properties. Another important work in this area of study is the Crystal Diffusion Variational Autoencoder (CDVAE) . By leveraging a diffusion process that pushes atomic coordinates to lower energy states and iterates atom types to satisfy bonding preferences, CDVAE significantly outperforms past attempts to perform material generation. Furthermore, CDVAE was capable of optimizing crystals for a given property and reconstructing a material from its latent space representation. Many examples exist for GANs as well. Constrained crystals deep convolutional generative adversarial network (CCDCGAN) uses a post-processing filtering criteria to remove compounds with large formation energies based on a cutoff threshold. In another example, crystal structure is represented in a lean fashion by a set of atomic coordinates and unit cell parameters, with significantly reduced memory requirements compared to a voxel-based generative model, iMatGen . Physics Guided Crystal Generative Model (PGCGM) expands on CubicGAN while using base site atoms with physics-guided loss functions. For example, structures with atoms that are too crowded or too far apart are penalized via two terms in the loss function. We also note that normalizing flows, another promising direction, have been used in some crystal structure contexts including an application to molecular crystals .