Ranking Pareto optimal solutions based on projection free energy

Based on available datasets prepared by numerical simulations and machine learning, maps of properties for materials that have not yet been synthesized can be developed. These maps can be used to select promising materials for synthetic experiments. With a single objective function, the ranking of the optimal solutions can be simply obtained based on the values of the target property. However, applications with multiple target properties require the calculation of Pareto optimal solutions to visualize trade-offs. These solutions are generally ranked manually, selecting the weight of the multiple objectives based on prior knowledge. In this study, to provide an automated ranking of Pareto solutions, we introduced the most-isolated Pareto solution (MIPS) score, which is defined by a projection free energy. Using the MIPS ranking, it is possible to appropriately select the most isolated materials predicted in the property space. To verify the effectiveness of the proposed method, we used a database of semiconductors created by density-functional theory. Our method was able to correctly select and rank the most isolated solutions in both convex and concave two-dimensional Pareto frontiers, outperforming the most relevant outlier detection methods. We also demonstrated that our approach can be easily extended to three-dimensional property spaces.

Figure 1

Schematics of Pareto solutions minimizing two objective functions and MIPS score. Based on the MIPS score, the ranking of Pareto solutions (orange) that are isolated from other datapoints are determined automatically.

Figure 2

(a) Examples of density of states for system A having few low-energy excited states and system B having many low-energy excited states. The temperature dependence on free energy results are shown for systems A and B. (b) Examples of two-dimensional Pareto frontier and projected one-dimensional DoS. For top and right DoS, objective functions 1 and 2 are only considered, respectively. These cases correspond to the Tchebycheff decomposition defined by Eq. (2) with $\alpha =1$ and 0, respectively. The Pareto solutions highlighted by red and green dotted circles correspond to the ground state (optimal solution) for top and right DoS cases, respectively. If a projection with different $\alpha$ is performed, different Pareto solution becomes optimal solution.

Figure 3

MIPS values (left column), representative DoS (middle column), and distribution of datapoints (right column) with a convex- or concave-shaped Pareto frontier in the property space spanned by the band gap and electric dielectric constant. The first 20 solutions ranked by MIPS are represented by blue datapoints. All DoS are summarized in Figs. S2 and S3 [28].

Figure 4

Distributions of outliers (orange) and other datapoints (gray) obtained using three types of outlier detection techniques (isolation forest, local outlier factor, and one-class SVM) when all the datapoints (left column) or the Pareto solutions only (right column) are used in the training phase.

Figure 5

MIPS values (left column), representative DoS (middle column), and distribution of datapoints (right column) with a three-dimensional Pareto frontier in the property space spanned by the band gap, and the electric and ionic dielectric constants. The first 20 solutions ranked by MIPS are represented by blue datapoints. All DoS are summarized in Fig. S4 [28]. Note that, since our model is developed to minimize objectives, when a problem requires a maximization of the target properties we simply consider their values with the opposite sign.