VAEs are based on autoencoders 101,102 , a class of unsupervised ML models which include an encoder to encode input data into a compressed representation in the latent space and a decoder to reconstruct the low-dimension representation to the representation of the original data as accurately as possible. En/decoder are typically composed of neural networks. Conceptually, the ultimate goal of autoencoders is to minimize the reconstruction loss so that the decoder can give output that closely resembles the original input. Autoencoders, while powerful for extracting latent representations, are trained to simply replicate the input rather than to learn how to generate truly new data from scratch. The emphasis of the model is reconstruction accuracy, and an extremely simplified situation is the model memorizes all training data so that the decoder exactly reproduces all original data without any reconstruction error. In this instance, no meaningful interpolation between the input and latent space is garnered and the model learns nothing. To really generate new data, several variants of autoencoders have be proposed. As the most popular variant of autoencoders, VAEs introduce variations into autoencoders. Plain autoencoders map the input into a fixed vector in the latent space, while VAEs map it into a distribution by sampling a prior distribution between the encoder and decoder, which allows for more flexible representation of the data. In addition, a new latent representation can be directly sampled from the latent space distribution and fed into the decoder and thus new data can be generated. Moreover, instead of consisting of distinct datapoints as in traditional autoencoders, the structured latent space distribution in VAEs allows for smooth interpolation between data points so that intermediate data points can be generated by interpolating between latent vectors. For example, given benzene and For inorganic crystal structures, iMatGen 107 constructs the latent space of solid-state materials by mapping the targeted compositions into image representation. To achieve this, iMatGen utilizes a two-step VAE for image reduction and material generation, respectively. A binary classifier is introduced to enhance the latent vector for the targeted formation energy related to the stability of materials. The model capably reproduced some experimental vanadium oxide materials as well as designing novel stable structures. Later, Court et al. extended iMatGen to design a VAE for not only generating new structures but also predicting eight properties of these generated structures simultaneously. In their work, the unit cells are embedded as electron-density maps and used to construct the latent space. To generate new structures, a UNet model is used to convert new electron-density maps sampled from the latent space to atomic structures. Finally, a graph convolutional neural network is used to predict the properties of generated structures. This conditional VAE model generates reasonable crystals with desired properties. However, the requirement of less than 40 atoms per unit cell limits the generation of complex structures. In addition, the model was trained on limited crystal-structure types, as a result, the model has limited generality and may not generate various crystal structures.