Fig. 8: ML structure representations and graph-based models for crystalline materials.

a A view of the FTCP matrix model that has both real and inverse space features. Real space includes properties such as CIF properties (element matrix - description of constituent elements, network matrix, description of network parameters conditions, site coordinate matrix - description of fractional coordinates of sites, site occupancy matrix - description of element occupancy at each site, and elemental attribute matrix - descriptors elemental). The cross-space features include the elemental descriptor Zi (i for each site) for all N locations in the unit cell along different spatial frequencies, hkl (Miller indices) via spatial discrete Fourier transform. The distance of each hkl (point k) from (000) to “dhkl” is also recorded in the cross-space features. In the variational autoencoder (VAE) architecture, using inverse FTCP representation to inverse design the encoder plus decoder architecture of a typical VAE, the latent space is also connected to a target learning branch for feature mapping, which reflects the feature gradients and the latent space structured with the corresponding feature. Reproduced with permission ref. ¹⁴³ Copyright 2021, Elsevier. b Map of a MEGNet module graph for the set of atomic attributes (V), bond attributes E, and global state attributes u. In the first update step, the bond attributes are updated. Information flows from atoms that form the bond, the state attributes, and the previous bond attribute to the new bond attributes. Similarly, the second and third steps update the atomic and global state attributes, respectively, by information flow among all three attributes. The final result is a new graph representation. Reproduced with permission ref. ¹⁴⁴ Copyright 2019, American Chemical Society. c Construction of the crystal graph, in this method crystals are converted to graphs with nodes representing atoms in the unit cell and edges representing atom connections. Nodes and edges are characterized by vectors corresponding to the atoms and bonds in the crystal, respectively. Structure of the CNN on top of the crystal graph (R convolutional layers, L₁, and L₂ hidden layers are built on top of each node, resulting in a new graph with each node representing the local environment of each atom), Reproduced with permission ref. ¹⁴⁵ Copyright 2018, American Physical Society.