Efficient construction method for phase diagrams using uncertainty sampling

We develop a method to efficiently construct phase diagrams using machine learning. Uncertainty sampling (US) in active learning is utilized to intensively sample around phase boundaries. Here, we demonstrate constructions of three known experimental phase diagrams by the US approach. Compared with random sampling, the US approach decreases the number of sampling points to about 20%. In particular, the reduction rate is pronounced in more complicated phase diagrams. Furthermore, we show that using the US approach, undetected new phases can be rapidly found, and smaller numbers of initial sampling points are sufficient. Thus, we conclude that the US approach is useful to construct complicated phase diagrams from scratch and will be an essential tool in materials science.

Figure 1

Overview of efficient phase diagram construction based on the uncertainty sampling approach.

Figure 2

Phase diagrams of ${\mathrm{H}}_2\mathrm{O}\text{−}\mathrm{L}$ (a), ${\mathrm{H}}_2\mathrm{O}\text{−}\mathrm{H}$ (b), and ${\mathrm{SiO}}_2\text{−}{\mathrm{Al}}_2{\mathrm{O}}_3\text{−}\mathrm{MgO}$ (c) and examples of samplings by using the US and RS approaches. Panels (a), (b), and (c) have 3, 6, and 10 phases, respectively. Panels (d), (e), and (f) show nine initial points randomly selected (triangles) and sampled points (circles) by the LP + LC method. Numbers of sampling points are 80, 110, and 180. Panels (g), (h), and (i) show sampling points by the RS method. Numbers of sampling points in panels (g), (h), and (i) are the same as in panels (d), (e), and (f). Relatively small phases such as ice III in panel (h) and tridymite and sapphirine in panel (i) are not found by the RS method.

Figure 3

Performances of phase diagram construction using the US approach. The top row shows the accuracies (Macro-F1) of phase diagram construction as functions of the number of sampling points for ${\mathrm{H}}_2\mathrm{O}\text{−}\mathrm{L}$ (a), ${\mathrm{H}}_2\mathrm{O}\text{−}\mathrm{H}$ (b), and ${\mathrm{SiO}}_2\text{−}{\mathrm{Al}}_2{\mathrm{O}}_3\text{−}\mathrm{MgO}$ (c). Colored lines are the results from the US approach. Black solid and dashed lines are from RS. The middle row shows the performances of phase detection. Panels (d), (e), and (f) show the number of sampling points necessary to detect all phases in the phase diagrams of ${\mathrm{H}}_2\mathrm{O}\text{−}\mathrm{L}$ (three phases), ${\mathrm{H}}_2\mathrm{O}\text{−}\mathrm{H}$ (six phases), and ${\mathrm{SiO}}_2\text{−}{\mathrm{Al}}_2{\mathrm{O}}_3\text{−}\mathrm{MgO}$ (ten phases). The bottom row shows the effects of the initial sampling points. Panels (g), (h), and (i) show the numbers of sampling points to reach a Macro-F1 score of 0.95 by using LP + LC as functions of initial sampling points (1, 4, 9, and 16).

Figure 4

Performances of the USPC approach for $\mathrm{H}{}_2\mathrm{O}–\mathrm{L}$ (a), $\mathrm{H}{}_2\mathrm{O}–\mathrm{H}$ (b), and $\mathrm{SiO}{}_2–\mathrm{Al}{}_2\mathrm{O}{}_3–\mathrm{MgO}$ (c). Colored lines are the results from the USPC approach. Black solid and dashed lines are from RSPC.