# Fileset [monte_carlo_tree_search_for_materials_design_and_discovery.pdf](https://mdr.nims.go.jp/filesets/a5d316cf-6a5e-4805-93ab-4cc287198450/download) ## Creator Ju, Shenghong, [Tsuda, Koji](https://orcid.org/0000-0002-4288-1606), Shiomi, Junichiro, [Dieb, Thaer M.](https://orcid.org/0000-0002-8111-2009) ## Rights ## Other metadata [Monte Carlo tree search for materials design and discovery](https://mdr.nims.go.jp/datasets/090e7531-770a-4683-b4da-54adb4795944) ## Fulltext S2159685919000405jrv 532..536Artificial Intelligence ProspectiveMonte Carlo tree search for materials design and discoveryThaer M. Dieb, National Institute for Materials Science, Tsukuba, Japan; Graduate School of Frontier Sciences, the University of Tokyo, Kashiwa, Japan;RIKEN, AIP, Tokyo, JapanShenghong Ju, National Institute for Materials Science, Tsukuba, Japan; Department of Mechanical Engineering, the University of Tokyo, Tokyo, JapanJunichiro Shiomi, National Institute for Materials Science, Tsukuba, Japan; Department of Mechanical Engineering, the University of Tokyo, Tokyo, Japan;CREST, JST, Tokyo, JapanKoji Tsuda, National Institute for Materials Science, Tsukuba, Japan; Graduate School of Frontier Sciences, the University of Tokyo, Kashiwa, Japan; RIKEN,AIP, Tokyo, JapanAddress all correspondence to Koji Tsuda at tsuda@k.u-tokyo.ac.jp(Received 15 January 2019; accepted 18 March 2019)AbstractMaterials design and discovery can be represented as selecting the optimal structure from a space of candidates that optimizes a target prop-erty. Since the number of candidates can be exponentially proportional to the structure determination variables, the optimal structure must beobtained efficiently. Recently, inspired by its success in the Go computer game, several approaches have applied Monte Carlo tree search(MCTS) to solve optimization problems in natural sciences including materials science. In this paper, we briefly reviewed applications ofMCTS in materials design and discovery, and analyzed its future potential.IntroductionThe ability to design a material with desired properties a prioriusing computational methods has been promised by computa-tional materials science for many years.[1] This problem canbe framed as selecting the optimal composite material structurethat meets certain quality metrics from a space of candi-dates.[2,3] One example is the structural determination of a sub-stitutional alloy problem in solid-state materials design,[4,5]where optimal atoms (or vacancies) assignment in a crystal struc-ture is determined to maximize or minimize a target property.To accelerate this process, researchers have emphasizeddata-driven and machine learning approaches as the fourth par-adigm of science.[6,7] Data-driven materials design approachesare iterative design algorithms. Given a space of candidates S,the algorithm aims to find the optimal candidate pbst that opti-mizes a black-box function f (p) (usually the target property).Starting with a random selection, and within a predefined num-ber of iterations, the algorithm evaluates a set of selected can-didates and obtains feedback for a more informed selection onthe next iteration. The function f (p) is evaluated by experimentor simulation and it is computationally expensive to query. It isnecessary to reach the optimal candidate with as few queries aspossible[8] (Fig. 1).Several methods have been applied for data-driven materialsdesign. Evolutionary algorithms, such as genetic algo-rithms[9,10] that use human evolution mechanisms (such ascrossover and mutation), have been used to solve optimizationproblems at a large scale. Genetic algorithms need adequatedata available a priori to tune many parameters for optimal per-formance. However, a priori data are often limited in materialsdesign and discovery.Another approach is Bayesian optimization (BO)[11,12] thatiteratively selects an optimal candidate from a search spacethat optimizes an expensive black-box function f (p). A deceiveadvantage of BO is that, it uses the predicated merit of the can-didate as well as the uncertainty of the prediction when select-ing the next promising candidate. This feature has beensuccessful in several materials design problems.[2,5,13–15]However, as the uncertainty of the prediction needs to beassessed for all candidates in the search space, excessive com-putation in BO faces serious challenges in large-scale prob-lems, a common case in materials design.The recent success of Monte Carlo tree search (MCTS)[16] incomputer games[17,18] has encouraged researchers to apply it tonatural science domains. For example, Yang et al. used MCTSfor de novo molecular generation by selecting an optimaldesign with a predefined desired criterion from a vast chemicalspace.[19] Segler et al. applied MCTS to discover retrosyntheticroutes to plan the syntheses of small organic molecules.[20]MCTS has also been applied in materials science and engi-neering. In this study, we reviewed the utilization of MCTS inmaterials design and discovery, particularly, to analyze its abil-ity to solve large-scale optimization problems. To the best ofour knowledge, no such review has been published yet. Thispaper is organized into four sections. The section “MonteCarlo tree search” presents the MCTS algorithm. In the sectionMRS Communications (2019), 9, 532–536© Materials Research Society, 2019. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creative-commons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.doi:10.1557/mrc.2019.40532▪ MRS COMMUNICATIONS • VOLUME 9 • ISSUE 2 • www.mrs.org/mrchttps://www.cambridge.org/core/terms. https://doi.org/10.1557/mrc.2019.40Downloaded from https://www.cambridge.org/core. National Institute of Materials Science (NIMS), on 02 Oct 2020 at 03:29:23, subject to the Cambridge Core terms of use, available atmailto:tsuda@k.u-tokyo.ac.jphttps://www.cambridge.org/core/termshttps://doi.org/10.1557/mrc.2019.40https://www.cambridge.org/core“Applications in materials design”, we discuss the utilization ofseveral variations of MCTS for finding optimal materialsdesign. Finally, “Concluding remarks” section summarizesthe paper with forward-looking remarks.Monte Carlo tree searchMCTS[16] is an iterative, guided, random best-first tree searchalgorithm that systemically searches a space of candidates toobtain an optimal solution that maximizes the black-boxfunction f (p). A distinguishing feature of MCTS is that itencodes the search space into a shallow tree that iterativelyexpands in the direction of the promising solutions, eliminat-ing the need to construct a full tree. MCTS then utilizesguided randomization to obtain full solutions from this shal-low tree.Within a computational budget, MCTS explores the searchspace over multiple iterations. Initially, only the root nodeexists. Then, each iteration consists of four steps: selection,expansion, simulation, and back-propagation (Fig. 2). In theselection step, the tree is traversed from the root to the mostpromising leaf using a comparative score. The chosen leaf isthen expanded by adding a child node to it in the expansionstep. Only partial solutions can be obtained from this shallowtree. In the simulation step, a full solution is generated by ran-dom rollout[16] starting from a node; the remaining variables arerandomly determined. The merit of the obtained solution isevaluated. In the back-propagation step, the nodes’ informationis updated along the path back to the root. MCTS balancesexploration against exploitation using a hyper parameter thatdepends on the range of maximum and minimum merits valuesof the candidates.Applications in materials designA Python package for an optimal atomassignment problemAn open source implementation of MCTS was developed byDieb et al. using the Python programing language.[21] Thepackage MDTS (materials design using tree search) is availableat https://github.com/tsudalab/MDTS. MDTS solves structuredetermination problems by optimal atom (or vacancy) assign-ment with composition constraints. This method assumes amaterial structure P with n positions {p1, p2, . . ., pn} thatneed to be assigned with several types of atoms (or vacancies).It searches for the optimal configuration pbst (subject to compo-sition constraints) that maximizes a black-box function f (p)(usually the target property). MDTS uses the MCTS model toencode the candidate space into a tree where a node at level lrepresents a possible atom assignment in the position l in thestructure. MDTS uses the upper confidence bound (UCB)score[16] to compare child nodes when traversing the tree.The UCB score is computed following Eq. (1):ucbi = zivi+ C�����������2 ln vparentvi√(1)where zi is the accumulated merit of the node (i.e., the sum ofthe immediate merits of all downstream nodes), vi is the visitcount of the node, vparent is the visit count of the parent node,and C is the constant to balance exploration and exploitation.MDTS adaptively sets C at each node following similaridea to[22]:C =��2√J4( fmax − fmin) (2)where J is a meta-parameter initially set to one, which increaseswhenever the algorithm encounters a dead-end leaf, to allow formore exploration. The parameters fmax and fmin are the maxi-mum and minimum immediate merits in the downstreamnodes, respectively. The automatic setting of C allows MDTSto work parameter-free with optimal performance on severalapplications. For example, MDTS successfully designed largesilicon–germanium (Si–Ge) alloy structures when BO couldnot be used due to its high computational cost.Grain boundary segregationAnalysis of the grain boundary segregation behavior of impuri-ties, dopants, and vacancies is very important for broad materi-als development due to its effect on material properties.Kiyohara et al.[23] recently applied the MCTS method to deter-mine a stable segregation configuration of copper Σ5[001]/(210) and Σ37[001]/(750) with silver impurities. A binaryMonte Carlo tree was constructed where a node representedeither a copper or silver atom assigned to a segregation site;the process searched for an optimum candidate with minimalsegregation energy. A stable copper Σ5[001]/(210) configura-tion was reached by searching only 1% of all candidateFigure 1. A data-driven materials design approach. Starting randomly, analgorithm selects a candidate set for experimentation within a computationalbudget. The experimental feedback is then used for a more informedselection in the next iteration.Artificial Intelligence ProspectiveMRS COMMUNICATIONS • VOLUME 9 • ISSUE 2 • www.mrs.org/mrc ▪ 533https://www.cambridge.org/core/terms. https://doi.org/10.1557/mrc.2019.40Downloaded from https://www.cambridge.org/core. National Institute of Materials Science (NIMS), on 02 Oct 2020 at 03:29:23, subject to the Cambridge Core terms of use, available athttps://github.com/tsudalab/MDTShttps://github.com/tsudalab/MDTShttps://www.cambridge.org/core/termshttps://doi.org/10.1557/mrc.2019.40https://www.cambridge.org/coreconfigurations (Fig. 3). This result was identical to a previousstudy using theoretical calculations, where empirical potentialsfor copper–silver alloys were generated to systematically inves-tigate the segregation.[24] The optimal solution of copper Σ37[001]/(750) (which had a significantly larger search space)was achieved after exploring only 0.34% of the search space.It is reported that the search efficiency was significantlyaffected by the order of the search (a search by order of segre-gation energy was more efficient than a random search).Additionally, the analysis of the search path and the numberof rollouts were important for understanding the backgroundof the search.Optimizing the energy gap for graphenenanoflakesThe study of the band gap in graphene is important for nanoe-lectronic applications. Cao et al. investigated the optimizationof graphene nanoflakes’ (GNFs) energy gaps by selecting theoptimal structure.[25] This study was challenged by the largenumber of candidate structures increasing as the number of car-bon atoms (Nc) in the graphene increased and the expensivecomputation of the energy gap (Eg) for a specific structure.The researchers proposed an effective tight-binding model forthe electronic properties of the hydrogen-terminated GNFsfocusing on the effect of carbon atom local bonding. Theparameters of this model were fit using first-principles calcula-tion results on structures with Nc≤ 34. For larger search spaces(Nc > 34), they used the MCTS method in combination with thecongruence check to search for larger GNFs structures withlarge Eg based on the properties of the smaller GNFs.Using this approach, they efficiently obtained optimal struc-tures with the largest Eg for each Nc. The reported results wereconfirmed with first-principles calculations.Surface/interface roughness optimizationMore recently, Ju et al.[26] employed MCTS to study the effectof surface/interface design in nanostructures for optimal ther-mal transport. They investigated the limits of inhibition andenhancement of phonon transport in the interfacial roughnessbetween a silicon–germanium (Si–Ge) bi-layer nanofilm(Fig. 4). In this setting, the number of possible roughnessFigure 2. MCTS encodes the search space as a shallow decision tree. MCTS repeats four steps. In the selection step, a promising leaf node is chosen byfollowing the child node with the best score. The expansion step adds a child node to the selected node. During simulation, a full solution is created by randomrollout and its merit is evaluated. The back-propagation step updates information for the nodes along the path back to the root for better selection in the nextiteration.Figure 3. (a) The promising configuration of the grain boundary structure ofcopper Σ5[001]/(210) up to the 16th site. Yellow and gray circles representsilver and copper atoms, respectively. (b) Strain map at each site at the grainboundary. The sites with a positive strain are larger spatially (red); the reverseholds for the negative strain (blue). Reprinted from Kiyohara andMizoguchi[23]; with the permission of AIP Publishing.534▪ MRS COMMUNICATIONS • VOLUME 9 • ISSUE 2 • www.mrs.org/mrchttps://www.cambridge.org/core/terms. https://doi.org/10.1557/mrc.2019.40Downloaded from https://www.cambridge.org/core. National Institute of Materials Science (NIMS), on 02 Oct 2020 at 03:29:23, subject to the Cambridge Core terms of use, available athttps://www.cambridge.org/core/termshttps://doi.org/10.1557/mrc.2019.40https://www.cambridge.org/coreconfigurations increased exponentially with the number of lay-ers in the nanofilm and the degree of roughness (e.g., there were1,048,576 candidates for 10 layers and four degrees).Researchers used the MDTS package[21] with the atomisticGreen’s function[27,28] to search for candidates with maximumand minimum interfacial thermal conductance. The reportedoptimal surface/interface roughness configurations were non-intuitive, and they were in the middle range between flat andvery rough. This study confirmed the scalability and efficiencyof MCTS because it applied MCTS to a large search space.Determination of boron-doping in a graphenestructureTuning material properties by incorporating additional ele-ments is a common practice in materials science and engineer-ing. Both optimal composition and spatial distribution ofincorporated atoms affect the target property. Dieb et al. inves-tigated the most stable structures of doped boron atoms in gra-phene (for a boron concentration up to 31.25%) using aMCTS-based method in conjunction with atomistic simula-tions.[29] They considered a supercell constructed of a hexago-nal unit cell of graphene and carbon substituted by boron.To increase the efficiency of MCTS, researchers engineeredthe rollout in the expansion step with a more sophisticated sol-ution using BO[11,12] (i.e., a Bayesian rollout). In this mecha-nism, a random pool of full candidates is enumerated underthe expanded node with a predefined size, Z. BO is then usediteratively to select the optimal candidate from the pool. BOmaintains a Gaussian process (GP)[30] as the surrogate modelof the objective function. As the MCTS progresses, moredata points are observed and included in the GP for training,which optimizes the model for the target function (Fig. 5).Using this method, researchers have reported atomic struc-tures of the most stable configurations of B–graphene at differentB concentrations. The stability of these structures has also beenverified by the density functional theory calculations with the useof the Vienna Ab initio Simulation Package (VASP).[31]Concluding remarksOur review of MCTS in materials design and discovery con-firmed that MCTS can successfully solve large-scale optimiza-tion problems in materials design. An interesting feature ofMCTS is that it does not require a complicated descriptor ora large dataset to tune many parameters. Instead, a singlehyper parameter can be tuned automatically and adaptivelybased on the target application. On the other hand, MCTScan be most useful when experimenting (simulation) time isshort. A significantly long experiment time will wipe out theadvantage of quick design time of MCTS. Additionally, currentimplementations of MCTS are only available for discretesearch spaces. Enhancing MCTS for continuous spaces cansupport a wider range of materials design applications.MCTS may have potential not yet fully exploited in materi-als science. A recent trend of combining MCTS with othermachine learning methods, such Bayesian learning and neuralnetworks, has emerged to increase MCTS’s efficiency, andthis has shown promising results.AcknowledgmentThis work was supported by the “Materials research byInformation Integration” Initiative (MI2I) project, the CRESTGrant No. JPMJCR16Q5 from the Japan Science andTechnology Agency (JST), and KAKENHI Grants No.16H04274 from the Japan Society for the Promotion ofScience (JSPS).References1. S.B. Sinnott: Material design and discovery with computational materialsscience. J. Vac. Sci. Technol. A 31, 050812 (2013).Figure 5. Bayesian rollout. A random pool of full candidates is generatedunder the expanded node. Initially, GP uses a random selection of data pointsfor evaluation. As the search progresses down the tree, more observationsare accumulated in the GP for a more informed future selection. To determinethe next selection, the Bayesian rollout uses an acquisition function thatconsiders the predicted value and prediction uncertainty.Figure 4. Interfacial roughness in a bi-layer nanofilm of Si–Ge. The numberof possible configurations increased exponentially with the number of layersand the degree of roughness.Artificial Intelligence ProspectiveMRS COMMUNICATIONS • VOLUME 9 • ISSUE 2 • www.mrs.org/mrc ▪ 535https://www.cambridge.org/core/terms. https://doi.org/10.1557/mrc.2019.40Downloaded from https://www.cambridge.org/core. National Institute of Materials Science (NIMS), on 02 Oct 2020 at 03:29:23, subject to the Cambridge Core terms of use, available athttps://www.cambridge.org/core/termshttps://doi.org/10.1557/mrc.2019.40https://www.cambridge.org/core2. A. Seko, A. Togo, H. Hayashi, K. Tsuda, L. Chaput, and I. Tanaka:Prediction of low-thermal-conductivity compounds with first-principlesanharmonic lattice-dynamics calculations and Bayesian optimization.Phys. Rev. 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Williams, eds.: Gaussian Processes forMachine Learning (MIT Press, Cambridge, MA, 2006).31.G. Kresse, and J. Furthmuller: Efficiency of ab-initio total energy calcula-tions for metals an semiconductors using a plane-wave basis set.Comput. Mater. Sci. 6, 15 (1996).536▪ MRS COMMUNICATIONS • VOLUME 9 • ISSUE 2 • www.mrs.org/mrchttps://www.cambridge.org/core/terms. https://doi.org/10.1557/mrc.2019.40Downloaded from https://www.cambridge.org/core. National Institute of Materials Science (NIMS), on 02 Oct 2020 at 03:29:23, subject to the Cambridge Core terms of use, available athttps://doi.org/10.1063/1.5023139https://doi.org/10.1063/1.5023139https://doi.org/10.1063/1.4961349https://doi.org/10.1063/1.4961349https://doi.org/10.1063/1.4961349https://doi.org/10.1063/1.5018065https://doi.org/10.1063/1.5018065https://www.cambridge.org/core/termshttps://doi.org/10.1557/mrc.2019.40https://www.cambridge.org/core Monte Carlo tree search for materials design and discovery Introduction Monte Carlo tree search Applications in materials design A Python package for an optimal atom assignment problem Grain boundary segregation Optimizing the energy gap for graphene nanoflakes Surface/interface roughness optimization Determination of boron-doping in a graphene structure Concluding remarks Acknowledgment References<< /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.3 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.1000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams true /MaxSubsetPct 100 /Optimize false /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage false /PreserveDICMYKValues true /PreserveEPSInfo false /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Remove /UCRandBGInfo /Remove /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 150 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /CHS /CHT /DAN /DEU /ESP /FRA /ITA /JPN /KOR /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. 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