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[Ryo Tamura](https://orcid.org/0000-0002-0349-358X), Kei Terayama, Masato Sumita, [Koji Tsuda](https://orcid.org/0000-0002-4288-1606)

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[Ranking Pareto optimal solutions based on projection free energy](https://mdr.nims.go.jp/datasets/43c56784-98e6-4715-9d51-4d7654d28892)

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Ranking Pareto optimal solutions based on projection free energyPHYSICAL REVIEW MATERIALS 7, 093804 (2023)Ranking Pareto optimal solutions based on projection free energyRyo Tamura ,1,2,3,4,* Kei Terayama ,5 Masato Sumita ,4,1 and Koji Tsuda 3,2,4,†1International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044, Japan2Research and Services Division of Materials Data and Integrated System, National Institute for Materials Science, Tsukuba 305-0044, Japan3Graduate School of Frontier Sciences, The University of Tokyo, Chiba 277-8561, Japan4RIKEN Center for Advanced Intelligence Project, Tokyo 103-0027, Japan5Graduate School of Medical Life Science, Yokohama City University, Kanagawa 230-0045, Japan(Received 22 February 2023; revised 22 June 2023; accepted 2 August 2023; published 19 September 2023)Based on available datasets prepared by numerical simulations and machine learning, maps of properties formaterials that have not yet been synthesized can be developed. These maps can be used to select promisingmaterials for synthetic experiments. With a single objective function, the ranking of the optimal solutions can besimply obtained based on the values of the target property. However, applications with multiple target propertiesrequire the calculation of Pareto optimal solutions to visualize trade-offs. These solutions are generally rankedmanually, selecting the weight of the multiple objectives based on prior knowledge. In this study, to providean automated ranking of Pareto solutions, we introduced the most-isolated Pareto solution (MIPS) score, whichis defined by a projection free energy. Using the MIPS ranking, it is possible to appropriately select the mostisolated materials predicted in the property space. To verify the effectiveness of the proposed method, we useda database of semiconductors created by density-functional theory. Our method was able to correctly select andrank the most isolated solutions in both convex and concave two-dimensional Pareto frontiers, outperformingthe most relevant outlier detection methods. We also demonstrated that our approach can be easily extended tothree-dimensional property spaces.DOI: 10.1103/PhysRevMaterials.7.093804I. INTRODUCTIONThe increasing computational power of high-performancecomputers and the development of automated techniques fornumerical simulations have facilitated the calculation of theproperties of a growing number of materials [1]. As a result,many accessible computational material databases have beendeveloped for inorganic materials [2,3], organic materials [4],and molecules [5,6]. In addition, the use of machine learn-ing in materials science has contributed to remarkable results[7–11]. By applying machine learning models to experimentaldata, the properties of unknown materials can be predictedwith high accuracy. Predicted properties can then be used as areference to select promising new materials, which will be thetarget of forthcoming experiments.If we consider a single target property, it is sufficient toselect the material with the highest (or lowest, depending onthe context) predicted property value. However, in the designprocess, materials are generally selected considering multipledesired properties at the same time. Moreover, the selection of*tamura.ryo@nims.go.jp†tsuda@k.u-tokyo.ac.jpPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.materials often leads to trade-offs (i.e., the enhancement of aproperty implies the worsening of another one). To treat suchtrade-off problems, multiobjective optimization methods canbe used [12–14]. Because a single solution does not exist forthis type of problem, these methods find multiple Pareto opti-mal solutions (see Fig. 1), each of which is an optimal solutionobtained by varying the balance of the objective functions.In general, the Pareto solutions are ranked using an arbitraryweighted sum [15–19]. The weights of the objectives areadjusted manually, thus requiring previous field knowledge.A systematic method for determining the balance of objectivefunctions was discussed in Refs. [20,21], where the centralpart of the Pareto frontier was regarded as that including thebest solutions. This means that the Pareto solutions are rankedbased on the proximity to the center of the frontier, whichis too simple. In general, without the direct use of humanknowledge, it is difficult to automatically select promisingmaterials when multiple objective properties are considered.In this study, we develop a method to automatically providethe ranking of Pareto solutions based on projection free en-ergy. Considering a material from the Pareto solutions, if thereare few others with similar properties in the property space(i.e., isolatability is high), then the material can be regardedas having unique properties (i.e., it is a curious material). Toevaluate the isolation of each Pareto solution, we introducethe most-isolated Pareto solution (MIPS) score, inspired bythe definition of free energy (see Fig. 1). Using the MIPSscore, the ranking of Pareto solutions is provided. Note thatthe ranking of the Pareto solutions considered in this study is2475-9953/2023/7(9)/093804(9) 093804-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-0349-358Xhttps://orcid.org/0000-0003-3914-248Xhttps://orcid.org/0000-0002-3506-1028https://orcid.org/0000-0002-4288-1606http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevMaterials.7.093804&domain=pdf&date_stamp=2023-09-19https://doi.org/10.1103/PhysRevMaterials.7.093804https://creativecommons.org/licenses/by/4.0/TAMURA, TERAYAMA, SUMITA, AND TSUDA PHYSICAL REVIEW MATERIALS 7, 093804 (2023)FIG. 1. Schematics of Pareto solutions minimizing two objectivefunctions and MIPS score. Based on the MIPS score, the ranking ofPareto solutions (orange) that are isolated from other datapoints aredetermined automatically.different from the Pareto ranking used by genetic algorithms[22]. In the Pareto ranking, all the solutions are ranked and allthe Pareto solutions are located at the highest rank.Let us explain the physical background of the idea that theisolation of each Pareto solution can be evaluated in terms offree energy. First, we consider two simple physical systemsA and B whose ground states are not degenerated, and theyhave the same number of discrete states. We also assume thatsystem A has few low-energy excited states and system Bhas many low-energy excited states. Examples of a densityof states (DoS) representing these systems are shown in theleft panels of Fig. 2(a). Focusing on the ground state, systemA has fewer low-energy excited states, so it can be assumedthat this ground state is more isolated from the other solutionsif the value of the energy is used as a measure of distance.System B, on the other hand, has many excited states in theneighborhood of the ground state, and the ground state canbe considered as nonisolated. Using the DoS, the temperaturedependence of the Helmholtz free energy can be calculatedfor each system and is shown in the right panel of Fig. 2(a).System A has a larger free energy, indicating that it is moreunstable than System B, and has an isolated ground state. Inthis way, ground state isolation can be measured in terms offree energy.Next, we consider the relationship between the isolation ofthe Pareto solution and the projection free energy. To apply theabove idea, it is necessary to consider a projection into a one-dimensional space from the space in which multiple objectivefunctions are defined. This projection is defined so that eachPareto solution becomes an optimal solution with the smallestvalue in the projected one-dimensional space, i.e., the groundstate. The simplest projection is to focus on only one of theobjective functions. For example, let us consider the datasetwith two objective functions as shown in Fig. 2(b). In thiscase, if we make a projection considering only one objectivefunction, the projected DoS is shown in Fig. 2(b). Note thatFIG. 2. (a) Examples of density of states for system A havingfew low-energy excited states and system B having many low-energyexcited states. The temperature dependence on free energy resultsare shown for systems A and B. (b) Examples of two-dimensionalPareto frontier and projected one-dimensional DoS. For top and rightDoS, objective functions 1 and 2 are only considered, respectively.These cases correspond to the Tchebycheff decomposition definedby Eq. (2) with α = 1 and 0, respectively. The Pareto solutionshighlighted by red and green dotted circles correspond to the groundstate (optimal solution) for top and right DoS cases, respectively. Ifa projection with different α is performed, different Pareto solutionbecomes optimal solution.these are the same DoS considered above, that is, for objectivefunctions 1 and 2, the DoS is the same as the systems A and B,respectively. For each projection, the Pareto solution that liesin the optimal solution is different. Thus, by estimating thefree energy in each case and comparing its magnitude [freeenergy results are shown in Fig. 2(a)], we can compare theisolation for different Pareto solutions. Here, the free energydefined in this projected space is called the projection freeenergy. In this example, via the projection free energy, we canconclude that the Pareto solution considering only objective093804-2RANKING PARETO OPTIMAL SOLUTIONS BASED ON … PHYSICAL REVIEW MATERIALS 7, 093804 (2023)function 1, highlighted by the red circle, is more isolatingthan the Pareto solution considering only objective function2, highlighted by green circle. For other Pareto solutions,we search for a one-dimensional space in which each Paretosolution becomes an optimal solution and calculate the pro-jection free energy. By comparing the projection free energy,the ranking of Pareto solutions based on the isolation canbe created. Here, the projection to one-dimensional space isperformed by Tchebycheff decomposition. Note that the rela-tionship between the difficulty in the optimization problemsand the shape of DoS has been discussed [23,24].A detailed definition based on projection free energy of theMIPS is provided in Sec. II. In Sec. III, we report the resultsof the material selection based on the MIPS using a dataset ofsemiconductors obtained by density functional theory (DFT).In Sec. IV, we present the discussion and summary of ourfindings.II. METHODSWe define the MIPS based on the concept of projection freeenergy. We consider a multiobjective optimization approachbased on the Tchebycheff decomposition [25–27] with twoobjective functions. Here, we assume that these are minimizedsimultaneously. It follows that the optimization problem in thed-dimensional space defined by x ∈ Rd can be described asfollows:minxgα (x), (1)where gα (x) is constructed using the two objective functionsf1(x) and f2(x) so thatgα (x) = max[α f1(x), (1 − α) f2(x)]. (2)The value of α (0 � α � 1) determines the balance of the twoobjectives. Note that the role of α is similar to that of thecoefficient of a weighted sum [i.e., α f1(x) + (1 − α) f2(x)].When α is small, f2(x) has more relevance than f1(x); whenα is large, the situation is reversed (see Fig. 1). When α isfixed, all data points are projected to one-dimensional datavia gα (x). Thus, a one-dimensional histogram can be drawndepending on α, and one of the Pareto solutions is located atthe optimal solution depending on α.For the analysis target of materials, the dataset of materialdescriptors D = {xi}i=1,...,N is given. Here, index i indicatesthe materials and D includes N types of materials. We con-sider the case in which ∀i the values of the objective functions{ f1(xi ), f2(xi )} are predicted by numerical simulations or ma-chine learning. Thus, in our problem setting, the materialsin the Pareto frontier can be easily determined. Note that amaterial with x∗ belonging to the Pareto optimal is one thatsatisfies the following condition: there is no xi for whichfk (xi ) < fk (x∗) for ∀k. We impose a min-max normalizationfor f1(x) and f2(x) defined as follows:f ′1(x) = f1(x) − min{ f1(xi )}i=1,...,Nmax{ f1(xi )}i=1,...,N − min{ f1(xi )}i=1,...,N, (3)f ′2(x) = f2(x) − min{ f2(xi )}i=1,...,Nmax{ f2(xi )}i=1,...,N − min{ f2(xi )}i=1,...,N. (4)The normalization ensures that the two objective functionshave the same range of values, thus contributing equally tothe target function that can be redefined as follows:g′α (x) = max[α f ′1(x), (1 − α) f ′2(x)]. (5)We perform an additional normalization so that the valuesof the target function are in the range [0, 1] for each α:Hα (x) = g′α (x) − min{g′α (xi )}i=1,...,Nmax{g′α (xi )}i=1,...,N − min{g′α (xi )}i=1,...,N. (6)Here, Hα (x) is regarded as the Hamiltonian function depend-ing on α from which we evaluate the projection free energy.The partition function depending on the temperature T isdefined as follows:Zα (T ) =N∑i=1exp [−Hα (xi )/T ]. (7)Thus, the projection free energy depending on α is given byFα (T ) = −T log Zα (T ) (8)= Eα (T ) − T Sα (T ), (9)where Eα (T ) and Sα (T ) represent the energy and entropyterms in the original definition of free energy. When T = 0,only the ground state with Hα (x) = 0 contributes to the freeenergy calculation and Fα (0) = 0. As T is increased, the con-tribution of excited states is augmented in Fα (T ). In particular,for small T , the contribution of the excited states close to theground state is important. When there is a limited number ofother data points near the ground state, the free energy Fα (T )with small T becomes large, corresponding to a low entropyvalue (see Fig. 2). Since the ground state is one of the Paretosolutions, we can use the free energy with small T when con-sidering the isolation from other solutions in the neighborhoodof the Pareto solution. Here, a sufficiently small temperatureT ∗ is predetermined, and the MIPS score is defined as follows:MIPS(α) = Fα (T ∗). (10)A Pareto solution with a large MIPS(α) can be regarded asisolated from the other data points. Note that, in this paper,T ∗ = 0.01 is utilized, and this value was determined by cal-culating the T ∗-dependence of the ranking (see Fig. S1 [28]),which will be discussed later.The following is a procedure to determine a ranking ofPareto solutions using the MIPS score.(1) For a given α, project all data into one-dimensionalspace.(2) Calculate the MIPS score according to Eq. (10). TheMIPS score represents the performance measure of the Paretosolution located in the optimal solution within the projectedone-dimensional space.(3) Repeat steps 1 and 2 while varying the value of α.In this case, the same Pareto solution may be the optimalsolution for different values of α. Therefore, each Pareto solu-tion can have multiple MIPS scores corresponding to differentα values. In such cases, the highest MIPS score among themultiple scores is chosen as the score for corresponding Paretosolution.(4) Arrange the Pareto solutions in ascending order oftheir MIPS scores.093804-3TAMURA, TERAYAMA, SUMITA, AND TSUDA PHYSICAL REVIEW MATERIALS 7, 093804 (2023)FIG. 3. MIPS values (left column), representative DoS (middle column), and distribution of datapoints (right column) with a convex- orconcave-shaped Pareto frontier in the property space spanned by the band gap and electric dielectric constant. The first 20 solutions ranked byMIPS are represented by blue datapoints. All DoS are summarized in Figs. S2 and S3 [28].However, in our present implementation, there might besome Pareto solutions for which the MIPS score cannot bedefined. Here, α is varied from 0 to 1 using discrete incre-ments, not continuous ones. Then, for considering all discreteα, there might be some Pareto solutions that do not become anoptimal (minimum) solution in the projected one-dimensionalspace. In such cases, these particular Pareto solutions shouldbe ranked as the lowest in the ranking.III. RESULTSA. MIPS ranking in a two-dimensional property spaceWe test our ranking model based on the MIPS us-ing a dataset of 1278 semiconductors obtained by densityfunctional theory (DFT) calculations [29]. We address thewell-known trade-off between the band gap and dielectricconstant of semiconductors. First, we consider the case inwhich both the band gap and electric dielectric constant (εel)are simultaneously minimized, which gives 33 Pareto solu-tions. The MIPS are calculated by varying α from 0 to 1 using0.001 increments. For seven Pareto solutions, the MIPS scorecannot be defined. In Fig. S1 [28], the transition of MIPSranking depending on T ∗ is shown. It can be seen that thePareto solutions in the top ranking do not change much whenT ∗ is changed from 0.001 and 0.01. Thus, we use T ∗ = 0.01,which achieves that the ranking does not change significantlyas decreasing T ∗ and includes sufficient information aroundPareto solution. Note that from T ∗ dependency, the robustnessof ranking can be discussed (see Fig. S1 [28]). In Table I, theMIPS ranking is summarized when T ∗ = 0.01, and for thematerials with 27th, an appropriate α cannot be found. The top20 results with the largest MIPS are shown in Fig. 3(a). Theleft column shows the MIPS values depending on the ranking.In the middle column, some representative DoS (first, tenth,and 20th in the ranking) are shown. Clearly, in the first DoS,the number of materials with low values in the projected one-dimensional space is small, indicating SiO2 is most isolated.In addition, we can see that the number of materials with lowvalues on the projected one-dimensional space increases aswe focus on the lower ranking. Thus, there are only a smallnumber of materials around a Pareto solution with a highMIPS. These results show that the ranking based on the MIPSis in accordance with our purpose of providing a ranking ofPareto solutions in terms of isolation. The distributions of theselected Pareto solutions and other datapoints in the propertyspace are shown in the right column of Fig. 3(a), where it canbe seen that a variety of isolated solutions were selected froma convex-shaped Pareto frontier using the MIPS ranking.We also consider a similar problem with a concave-shapedPareto frontier. In this case, the band gap and electric dielectric093804-4RANKING PARETO OPTIMAL SOLUTIONS BASED ON … PHYSICAL REVIEW MATERIALS 7, 093804 (2023)TABLE I. MIPS ranking for a convex-shaped Pareto frontierwhen band gap and electronic dielectric constant (εel) are minimized.Ranking Materials MIPS score Band gap εel1 SiO2 −0.002270662 5.79 2.22 RbMnO4 −0.002309888 2.02 2.93 K3BiO4 −0.004522329 1.24 3.34 Na6PbO5 −0.007955985 1.1 3.65 Rb2Be2O3 −0.00853923 2.42 2.86 K4PbO4 −0.009820247 1.68 3.17 Na2Si2O5 −0.012068143 4.54 2.38 K4SnO4 −0.012493567 2.35 2.99 K2Na4Be2O5 −0.013082062 2.53 2.810 CsLi(B3O5)2 −0.013750656 5.15 2.311 KNa2BO3 −0.013784018 2.98 2.512 K3AuO −0.014702593 0.5 4.913 Na7Al3O8 −0.014925278 2.91 2.614 Na3BO3 −0.015025572 3.15 2.415 Na5BiO5 −0.015135497 1.03 3.916 K4HfO4 −0.01615541 2.78 2.717 Na4SiO4 −0.018832454 3.4 2.418 Cd2Sb2O7 −0.019424779 0.54 4.819 Rb5Au3O2 −0.019745257 0.9 4.120 Na4GeO4 −0.021150197 2.89 2.721 Na2SiO3 −0.023911928 3.99 2.422 KBO2 −0.024760391 4.12 2.423 Rb7Au5O2 −0.024863074 0.68 4.724 NaBO2 −0.025005608 4.19 2.425 NaCuO2 −0.025564282 0.79 4.626 SnO2 −0.026644142 0.86 4.527 AlAsO4 - 4.39 2.427 K4GeO4 - 2.78 2.827 Na4B2O5 - 3.92 2.427 Rb3NaPbO4 - 1.63 3.327 Na3AsO4 - 3.38 2.427 Rb4PbO4 - 1.6 3.327 CsCdBO3 - 1.69 3.1constant are simultaneously maximized and 31 Pareto solu-tions are defined. Since the definition of the MIPS is basedon the minimization of the objective functions, we considerthe corresponding negative values of the two properties of thedielectric materials, so that the original maximization is trans-formed into a minimization problem that can be addressed bythe proposed method. In this case, for four Pareto solutions,MIPS score cannot be defined. In Table II, the MIPS ranking issummarized when T ∗ = 0.01, and the results are presented inFig. 3(b). Note that in Fig. S1 [28], MIPS ranking dependingon T ∗ is shown. The first two materials (BeO and PtO2) withthe highest rank appear highly isolated. This result is correctlyreflected by the associated MIPSs. We conclude that the MIPSranking can be also used for problems with a concave-shapedPareto frontier.B. Comparison with outlier detection techniquesOur ranking method can be used to extract isolated dat-apoints from Pareto solutions. Outlier detection techniquesare used to perform the same task using machine learning.To investigate the possible advantages of our ranking method,TABLE II. MIPS ranking for a concave-shaped Pareto frontierwhen band gap and electronic dielectric constant (εel) are maximized.Ranking Materials MIPS score Band gap εel1 BeO −1.48E-09 7.47 3.12 PtO2 −3.91E-07 0.95 12.53 ThO2 −0.00065343 4.45 4.84 HfSiO4 −0.000891161 5.66 3.75 In2Pt2O7 −0.001298629 2.48 6.96 ScTaO4 −0.00144392 4.08 57 ZrSiO4 −0.002396839 5.06 48 TaBiO4 −0.002783542 3.02 6.19 CaHfO3 −0.002949965 4.98 4.110 LaScO3 −0.003333902 3.87 5.111 V2HgO6 −0.003943528 2.09 7.312 ScNbO4 −0.00517729 3.53 5.513 Ta2Bi4O11 −0.005263738 2.77 6.214 TlPd3O4 −0.006784382 1.76 9.815 CaZrO3 −0.007067831 4.47 4.416 BeAl2O4 −0.007128697 6.27 3.217 Sc2Pt2O7 −0.007141839 2.24 718 Al2O3 −0.007150093 6.04 3.319 NaNb4Bi5O18 −0.007385994 2.49 6.420 TiO2 −0.007594438 2.94 6.121 Nb2CdO6 −0.007769357 3.54 5.422 TiCdO3 −0.007860211 3.03 5.823 SrHfO3 −0.008652802 4.68 4.124 TlPt3O4 −0.009422992 1.78 9.625 LaTaTiO6 −0.009535061 3.08 5.626 TiPbO3 −0.009952586 1.84 7.927 Bi2O3 −0.011299502 2.15 728 YZnAsO - 1.11 9.828 CaTa2Bi2O9 - 3.03 5.628 Ta4PbO11 - 3.5 5.528 Ta2CdO6 - 4.08 4.8we apply the outlier detection techniques, namely, isolationforest [30], local outlier factor [31], and one-class supportvector machine (SVM) to solve the maximization problem ofband gap and electronic dielectric constant. The results arepresented in Fig. 4. We consider the following two cases: (1)all data, including Pareto solutions, are used as the trainingdataset; (2) only Pareto solutions are used as the trainingdataset. Considering case (1), the isolation forest and one-class SVM recognize a datapoint/solution as an outlier whenat least one of the objective functions is particularly small.This means that these techniques cannot be used to definethe isolated Pareto solutions. Because the local outlier factorcorrectly selects all the Pareto solutions, it is not suitable forthe ranking process. Considering case (2), only the solutionslocated at the extremities of the Pareto frontier are selectedas outliers by the considered methods. These results indicatethat it is impossible to create the ranking of Pareto solutionsby using outlier detection techniques.C. MIPS ranking in a three-dimensional property spaceBy extending the target function domain, we can apply ourmethod to a property space of more than two dimensions. In093804-5TAMURA, TERAYAMA, SUMITA, AND TSUDA PHYSICAL REVIEW MATERIALS 7, 093804 (2023)FIG. 4. Distributions of outliers (orange) and other datapoints(gray) obtained using three types of outlier detection techniques (iso-lation forest, local outlier factor, and one-class SVM) when all thedatapoints (left column) or the Pareto solutions only (right column)are used in the training phase.particular, in this section, we address the three-dimensionalcase. In addition to the band gap and the electronic dielectricconstant, we consider the ionic dielectric constant as the thirdproperty. We also consider the case in which these propertiesare maximized and 70 Pareto solutions are determined. Here,the target function inspired by the Tchebycheff decompositionis as follows:gα (x) = max[α1 f1(x), α2 f2(x), α3 f3(x)], (11)where f1(x), f2(x), and f3(x) are the three objective func-tions. Choosing the values of α1, α2, and α3, we apply twoconstraints: αi > 0 (i = 1, 2, 3) and α1 + α2 + α3 = 1. Underthese constraints, the MIPS are calculated by varying αi from0 to 1 using a 0.01 increments. In this case, for 15 Paretosolutions, the MIPS score cannot be defined. The top 20results with the largest MIPS are shown in Table III and Fig. 5.From the distributions, it can be observed that a wide varietyof Pareto solutions are chosen.TABLE III. MIPS ranking of top 20 for a three-dimensionalPareto frontier when the band gap, and the electric (εel) and ionic(εion) dielectric constants are maximized. The remaining ranking isshown in Table S1 [28].Ranking Materials MIPS score Band gap εel εion1 BeO −1.48E-09 7.47 3.1 3.92 PtO2 −3.91E-07 0.95 12.5 1.23 TiPbO3 −1.05E-06 1.84 7.9 661.44 La3TaO7 −7.48E-05 3.63 4.8 73.45 Sr4Ta2O9 −0.000120481 4.4 3.7 32.76 CaHfO3 −0.000177427 4.98 4.1 17.27 Sr4As2O −0.000247042 1.06 8.3 62.88 CaZrO3 −0.000429054 4.47 4.4 22.59 Rb2Ti(WO4)3 −0.000444705 2.83 5.3 98.310 TaBiO4 −0.000463315 3.02 6.1 3611 Ta4PbO11 −0.000495176 3.5 5.5 27.212 ThO2 −0.00065306 4.45 4.8 11.913 HfSiO4 −0.00089102 5.66 3.7 714 NaNb4Bi5O18 −0.000948134 2.49 6.4 48.215 Bi2O3 −0.001084116 2.15 7 24.916 TlPd3O4 −0.001142092 1.76 9.8 25.617 La3HfGa5O14 −0.00128972 3.8 4.1 53.418 In2Pt2O7 −0.00130258 2.48 6.9 5.419 ScTaO4 −0.001393008 4.08 5 16.920 Tl4O3 −0.001521282 0.72 10.2 32.5High-k dielectrics are materials that have wide band gapsand high dielectric constants [32,33]. For example, theseproperties are crucial for gate dielectrics and capacitors insemiconductor technology [34]. In this section, we focus ona Pareto frontier to maximize both the band gap and dielec-tric constants. This helps in searching for suitable materialsfor high-k dielectrics. Among the top ten materials in theMIPS ranking, there are some materials that have not beenthoroughly investigated through experiments. These materi-als include Sr4Ta2O9, CaHfO3, Sr4As2O, Rb2Ti(WO4)3, andTaBiO4. The MIPS ranking suggests that it would be in-teresting to experimentally verify these materials. Sr4Ta2O9and CaHfO3 are particularly promising materials as they pos-sess wide band gaps and high dielectric constants and someprevious studies were conducted on these materials [35,36].Rb2Ti(WO4)3 and TaBiO4 are also considered promisingbecause they exhibit significantly high dielectric constantsand their band gaps are comparable to those of BaTiO3 andSrTiO3, which are commonly used in ceramic capacitor appli-cations [37,38]. On the other hand, Sr4As2O has a narrowerband gap but a larger dielectric constant. Its inclusion in theMIPS ranking adds diversity to the material choices. Ideally,it would be beneficial to investigate all the materials that arePareto solutions, but in this case, there are 70 Pareto solutions.The MIPS ranking provides a strategy to prioritize the orderin which these materials should be considered. In recent years,the amount of data in computational materials databases haveincreased drastically. Consequently, the number of Paretosolutions in such databases is also expected to increase.By utilizing these databases, the MIPS ranking is anticipatedto effectively facilitate the exploration of new materials.093804-6RANKING PARETO OPTIMAL SOLUTIONS BASED ON … PHYSICAL REVIEW MATERIALS 7, 093804 (2023)FIG. 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 solutionsranked by MIPS are represented by blue datapoints. All DoS are summarized in Fig. S4 [28]. Note that, since our model is developed tominimize objectives, when a problem requires a maximization of the target properties we simply consider their values with the opposite sign.IV. DISCUSSION AND SUMMARYWe proposed a method to automatically provide the rank-ing of Pareto solutions in order of decreasing isolation. Tomeasure the isolation of each Pareto solution, we definedthe most-isolated Pareto solution (MIPS), which was definedby the free energy. We used a multiobjective optimizationapproach based on Tchebycheff decomposition to projectmultiple objective functions to one-dimensional function. Totest the proposed method, we used a dataset of semiconduc-tor and a two- and three-dimensional property spaces wereconsidered. Our results show that the most-isolated Paretosolutions can be selected automatically using our ranking. Theimplementation of our method can be found on GitHub [39] inthe PYTHON package called Ranking of Pareto solutions basedon Projection Free energy (RPPF).The ranking of Pareto solutions based on the MIPS canbe used in several applications, for instance, in the anal-ysis of simulation results and in association with existingoptimization methods. Numerical simulations for materialsdesign can be performed using large datasets of materials,resulting in several Pareto solutions. Using a ranking methodfor these Pareto solutions, materials with interesting prop-erties can be selected among them. Subsequent research onthe selected solutions may deepen our understanding of theunusual properties of materials and contribute to the estab-lishment of guiding principles for the development of newones.Moreover, our method can be used in cooperation withblack-box optimization [40], which uses machine-learningpredictions to select appropriate materials that have not yetbeen synthesized or simulated [41–44]. Using the MIPS rank-ing, it would be possible to select a wide variety of materialsfrom Pareto solutions predicted by machine learning. The se-lected materials will be evaluated by conducting experimentsor simulations. Repeating the selection and evaluation pro-cesses of the black-box optimization, desired materials willbe discovered. Using the MIPS would prevent the selectionof materials with similar properties. This could improve theperformance of batch experiments where robotic systems formaterials development are employed [45–49].In this study, we focused only on trade-off problems ofsemiconductors. However, several disciplines, including ma-terials science, physics, and chemistry, aim at solving this typeof problem. Some relevant examples of trade-offs that couldbe addressed by the MIPS ranking based on the projection freeenergy are the efficiency and power trade-off in thermody-namic systems [50], the permeability and selectivity trade-offcharacterizing polymer membranes [51], and the cost andperformance trade-off in scientific research.ACKNOWLEDGMENTSWe thank Akira Takahashi, Fumiyasu Oba, Yohei Aki-moto, and Koji Hukushima for the useful discussions.Numerical calculations were performed using a supercom-puter made available by the Institute for Solid State Physics,University of Tokyo. 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