Fig. 1: Schematics of the method for phase-field ranking for synthetic exploration.

Stage I: ML model training from ICSD data of reported chemical systems in four steps: 1 quaternary phases with two anions are collected, including, for example, NaV₂O₄F.; 2 The individual phases (e.g., both the NaV₂O₄F and Na₂VOF₅) are aggregated to define those phase fields that contain reported two anions quaternaries. Each phase field is represented as a 148-dimensional (four elements × 37 elemental features) vector p; 3 the VAE²⁹ is used to reduce (encode) p into a four-dimensional latent vector, \(\tilde{{{{{{\bf{p}}}}}}}\); 4 the VAE decodes \(\tilde{{{{{{\bf{p}}}}}}}\) back to a 148-dimensional image vector, p′. During training the VAE tunes the weights and biases of its neural networks to minimize the reconstruction errors—the Euclidean distances between the original, p, and decoded, p′, vectors. When exploring a new quaternary phase field with two anions (Stage II), e.g., Li-La-O-Br, we use the trained VAE model to encode the new phase field into the latent space and then decode it while measuring its reconstruction error. The reconstruction error of a candidate phase field captures the degree of its deviation from reported chemical systems, and enables the ranking of unexplored phase fields for synthetic exploration.