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.