Application of machine learning potentials to predict grain boundary properties in fcc elemental metals
AApplication of machine learning potentials to predict grain boundary propertiesin fcc elemental metals
Takayuki Nishiyama, ∗ Atsuto Seko, † and Isao Tanaka
1, 2, 3 Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan Center for Elements Strategy Initiative for Structure Materials (ESISM), Kyoto University, Kyoto 606-8501, Japan Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan (Dated: August 14, 2020)Accurate interatomic potentials are in high demand for large-scale atomistic simulations ofmaterials that are prohibitively expensive by density functional theory (DFT) calculation. In thisstudy, we apply machine learning potentials in a recently constructed repository to the predictionof the grain boundary energy in face-centered-cubic elemental metals, i.e., Ag, Al, Au, Cu, Pd, andPt. The systematic application of machine learning potentials shows that they enable us to predictgrain boundary structures and their energies accurately. The grain boundary energies predicted bythe MLPs are in agreement with those calculated by DFT, although no grain boundary structureswere included in training datasets of the present MLPs.
I. INTRODUCTION
Grain boundaries are interfaces between differentlyoriented crystals of the same phase [1]. The microstruc-tures of grain boundaries can affect various properties ofpolycrystalline materials, including mechanical, thermal,and electrical properties [2–5]. Thus, an attractive topicin materials science has been to establish the relationshipbetween the properties of crystalline materials and grainboundary structures. Many theoretical studies have beenmade to cover a broad range of grain boundary structuresand their excessive energies. Early fundamental studiesemployed pair potentials, such as the Lennard–Jonesand Morse forms, to investigate the generic propertiesof grain boundaries such as the presence of cusps in amap of the rotation angle and the grain boundary energy[6–8]. Empirical interatomic potentials such as theFinnis–Sinclair (FS) potentials [9] and embedded atommethod (EAM) [10] potentials have been widely usedto investigate symmetric and asymmetric grain bound-aries of metallic materials. Quantitative predictionsare becoming possible [11–20], and strong correlationsbetween theoretical and experimental grain boundaryenergies have been shown, especially for grain boundariesin elemental Al and Ni, which exhibit low grain boundaryenergies [21, 22]. However, the prediction error inthe grain boundary energy may be significant in grainboundaries showing higher grain boundary energies. Thiserror originates from the fact that their microscopic grainboundary structures differ from the atomic environmentused to estimate interatomic potentials.Density functional theory (DFT) calculation [23, 24]is an alternative way to predict grain boundary prop-erties accurately. However, DFT calculation is practi-cally impossible to apply to large-scale models of grainboundaries owing to its computational cost. Therefore, ∗ [email protected] † [email protected] interatomic potentials that enable us to predict grainboundary properties accurately have been in high de-mand. Over the last decade, many groups have pro-posed frameworks to develop machine learning potentials(MLPs) based on extensive datasets generated by DFTcalculation [25–46]. The MLPs significantly improve theaccuracy and transferability of interatomic potentials.Also, MLPs themselves are becoming available, suchas those in Machine Learning Potential Reposi-tory [47] developed by one author of this paper.In this paper, we demonstrate the predictive powerof MLPs in the MLP repository for grain boundaryproperties. We systematically evaluate the structuresand excessive energies of (cid:104) (cid:105) symmetric tilt grainboundaries (STGBs), (cid:104) (cid:105)
STGBs, and (cid:104) (cid:105) pure-twist grain boundaries in the face-centered-cubic (fcc)elemental metals of Ag, Al, Au, Cu, Pd, and Pt. Theyare compared with those obtained from EAM potentialsand DFT calculations. The MLP repository containsa set of Pareto optimal MLPs with different trade-offsbetween accuracy and computational efficiency; hence,we carefully determine appropriate MLPs to predict grainboundary properties.
II. METHODOLOGYA. Modeling and structure optimization of grainboundaries
Macroscopic structures of grain boundaries are charac-terized by five geometrical degrees of freedom. We choosethree variables to specify the direction of the rotation axisand the rotation angle, which describe the misorientationbetween crystal lattices, and two variables to specify thedirection of the boundary plane normal [1]. For a givenset of macroscopic variables, the microscopic structureis associated with three degrees of freedom regardingrigid body displacements: two components parallel tothe boundary plane and one component normal to the a r X i v : . [ phy s i c s . c o m p - ph ] A ug plane. Hence, the globally optimal microscopic structurefor a given set of macroscopic variables is achieved byoptimizing the three microscopic variables in terms ofpotential energy.In this study, we investigate only STGBs and pure-twist grain boundaries. The periodicity of an STGBis identified from the orthogonal projection of its coin-cident site lattice (CSL) to its boundary plane. Also,the periodicity of a pure-twist grain boundary is givenby the orthogonal projection of its displacement shiftcomplete (DSC) lattice to its boundary plane. Therefore,we restrict the ranges of the two in-plane microscopicvariables to a domain defined by the periodicity of thegrain boundaries.We explore the globally optimal microscopic structurefor a set of macroscopic variables using a multi-startmethod. The multi-start method involves local structureoptimizations for a given set of initial structures andregards the structure with the lowest energy amongthe converged final structures as the globally optimalstructure. We use the conjugate gradient methodimplemented in the lammps code [48] for the localstructure optimizations. Initial microscopic structuresare introduced from a 4 × pymatgen [49]. This model contains two parallelboundaries perpendicular to the c-axis of the model, sep-arated by fcc layers corresponding to four repetitions of acell of the CSL. However, the local structure optimizationstarting from some of the initial microscopic structuresfails to converge when using both the MLPs and theEAM potentials, as shown in the next section. Thesestructures are ignored in finding the globally optimalmicroscopic structure. Note that the optimization of themicroscopic structure is performed in the whole domainhere, although it is more efficient to restrict the domainto its symmetrically nonequivalent domain. B. Machine learning potentials
We employ MLPs in
Machine Learning PotentialRepository [47] developed by one author of this paperto obtain the globally optimal microscopic structuresof STGBs and pure-twist grain boundaries. In therepository, a set of Pareto optimal MLPs with differenttrade-offs between accuracy and computational efficiencyis available, from which one can choose an appropriateMLP in accordance with the target and purpose. Poten-tial energy models of the MLPs are either a polynomialmodel of Gaussian-type pairwise structural features or apolynomial model of polynomial invariants for the O(3)group, which are derived by a group-theoretical approach[50].The Pareto optimal MLPs in the repository havebeen developed using a dataset generated from struc- ture generators. For Ag, Al, Au, and Cu, we adoptthe Pareto optimal MLPs developed from a structuregenerator set composed of the fcc, body-centered-cubic(bcc), hexagonal-close-packed (hcp), simple cubic (sc), ω , and β tin structures. The dataset is composedof 3,000 structures constructed by introducing randomlattice expansion, random lattice distortion, and randomatomic displacements into a supercell of the equilibriumstructure for one of the structure generators. For Pd andPt, we employ another set of 82 prototype structures asthe structure generator set because the dataset derivedfrom the six structure generators is not available in therepository. The dataset consists of 10,000 structuresgenerated by the same procedure as above. For allstructures in the dataset, DFT calculations were per-formed using the plane-wave-basis projector augmentedwave method [51] within the Perdew–Burke–Ernzerhofexchange-correlation functional [52] as implemented inthe VASP code [53–55]. Note that the datasets containno structures generated from grain boundary models.
III. RESULTS AND DISCUSSION
First, we systematically calculate the grain boundaryenergies of five grain boundaries using the whole set ofPareto optimal MLPs for each elemental metal. Theyare the Σ5 (cid:104) (cid:105) STGB (at 53.1 degrees), the Σ3 (cid:104) (cid:105) STGB (at 70.5 degrees), the Σ3 (cid:104) (cid:105) STGB (at 109.5degrees), the Σ9 (cid:104) (cid:105) STGB (at 38.9 degrees), and the Σ5 (cid:104) (cid:105) pure-twist grain boundary (at 36.9 degrees),the calculation models for which can be represented bya small number of atoms. Then, we find an accurateMLP requiring only a reasonable computational time toinvestigate the whole set of grain boundaries.Figure 1 shows the convergence behavior of the grainboundary energy in terms of the computational time, ob-tained using the whole set of Pareto optimal MLPs. Thegrain boundary energy is identical to the lowest energyamong the grain boundary energies of the microscopicstructures. The grain boundary energy of a microscopicstructure is measured from the energy of the equilibriumfcc structure. The computational time corresponding tothe model complexity of an MLP is the elapsed timenormalized by the number of atoms for a single pointcalculation of the energy, the forces, and the stress ten-sors [56]. As can be seen in Figure 1, the grain boundaryenergy converges well in all of the elemental metals andgrain boundaries. Consequently, successive calculationsfor the whole set of grain boundaries are performed usingthe MLP that requires the lowest computational timeamong the MLPs showing convergence.We also examine the transferability of the MLPs tothe prediction of the grain boundary structures andenergies because the datasets used in developing theMLPs contain no grain boundary structures. Therefore,we evaluate the grain boundary energies of the Σ3 (cid:104) (cid:105) STGB (at 70.5 degrees), the Σ3 (cid:104) (cid:105) STGB (at 109.5 (cid:104) (cid:105)
STGB (cid:104) (cid:105)
STGB (cid:104) (cid:105) pure-twist grain boundaryAg -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Ag <100>STGB Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Ag <110>STGB Σ °Σ °Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Ag <100>pure-twist Σ ° Al -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Al <100>STGB Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Al <110>STGB Σ °Σ °Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Al <100>pure-twist Σ ° Au -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Au <100>STGB Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Au <110>STGB Σ °Σ °Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Au <100>pure-twist Σ ° Cu -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Cu <100>STGB Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Cu <110>STGB Σ °Σ °Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Cu <100>pure-twist Σ ° Pd -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Pd <100>STGB Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Pd <110>STGB Σ °Σ °Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Pd <100>pure-twist Σ ° Pt -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Pt <100>STGB Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Pt <110>STGB Σ °Σ °Σ ° -5 -4 -3 -2 -1 G r a i n B ounda r y E ne r g y [ m J / m ] Elapsed Time [s/atom/step] (Single CPU Core)Pt <100>pure-twist Σ ° Figure 1. Grain boundary energies of Σ5 (cid:104) (cid:105) STGB in 53.1 degrees, Σ3 (cid:104) (cid:105) STGB in 70.5 degrees, Σ3 (cid:104) (cid:105) STGB in109.5 degrees, Σ9 (cid:104) (cid:105) STGB in 38.9 degrees, and Σ5 (cid:104) (cid:105) pure-twist grain boundary in 36.9 degrees for elemental Ag, Al,Au, Cu, Pd, and Pt, predicted using the Pareto optimal MLPs. The grain boundary energies computed by DFT calculationare also shown by broken lines. degrees), the Σ9 (cid:104) (cid:105) STGB (at 38.9 degrees), the Σ5 (cid:104) (cid:105) STGB (at 53.1 degrees), and the Σ5 (cid:104) (cid:105) pure-twist grain boundary (at 36.9 degrees) by DFT cal-culation, and compare them with those predicted usingthe MLPs. Figure 1 shows the DFT values of the grainboundary energy only for the grain boundary structures,DFT calculations for which converge successfully. Theyare close to the grain boundary energies of the selectedMLPs. Therefore, the selected MLPs should have highpredictive power for grain boundary structures and theirenergies even though no grain boundary structures wereused to develop the MLPs.Table I lists the model parameters of the selectedMLPs. Fast MLPs are selected for Ag and Cu, while com-putationally expensive MLPs are selected for the others.Table I also shows the prediction errors for the datasetsused in developing the MLPs. The MLPs for Pd and Ptshow significant prediction errors, which originate fromthe fact that the datasets contain many hypotheticalstructures such as the graphite-type structure. Althoughthe selected MLPs exhibit significant prediction errorsfor such abnormal structures, they show much smallerprediction errors for typical metallic structures, includinggrain boundary structures, as shown above.After confirming the transferability of the MLPs, wecalculate the energies of the grain boundary structures: (cid:104) (cid:105) STGBs ( Σ5 , Σ13 , Σ17 , Σ25 , Σ29 , Σ41 ), (cid:104) (cid:105) STGBs ( Σ3 , Σ9 , Σ11 , Σ17 , Σ19 , Σ27 , Σ33 , Σ41 , Σ43 ),and (cid:104) (cid:105) pure-twist grain boundaries ( Σ5 , Σ13 , Σ17 , Σ25 , Σ29 , Σ37 , Σ41 ). Most of them are representedby large-scale models, hence they cannot be calculatedby DFT calculation because of the large computationalresources required. Figure 2 shows the rotation angledependence of the grain boundary energy obtained usingthe MLPs and EAM potentials [57–61]. The values of thegrain boundary energy in Al, Cu, and Pd computed usingthe MLPs are consistent with those computed using theEAM potentials and those computed by DFT calculation.Therefore, both the MLPs and the EAM potentials havehigh predictive power for the grain boundary structures and their energies. In Ag, Au, and Pt, the values of thegrain boundary energy computed using the MLPs arealmost the same as those computed by DFT calculation,whereas they deviate from those computed using theEAM potentials. The MLPs should be more reliablethan the EAM potentials for obtaining not only thegrain boundary structures and their energies but alsothe other defect structures in Ag, Au, and Pt. Notethat a fine sequence is required for the component normalto the boundary plane to obtain converged microscopicstructures when using the EAM potentials for Ag andAu. This implies that the EAM potentials for Ag and Aulack accuracy for predicting the potential energy surfacearound the globally optimal microscopic structure.
IV. CONCLUSION
We have examined the predictive power of MLPs inan MLP repository for grain boundary properties bysystematically evaluating the grain boundary energy for (cid:104) (cid:105)
STGBs, (cid:104) (cid:105)
STGBs, and (cid:104) (cid:105) pure-twist grainboundaries in the fcc elemental metals of Ag, Al, Au,Cu, Pd, and Pt. In every elemental metal, the valuesof the grain boundary energy computed using the MLPare consistent with those computed by DFT calculation.We emphasize that the training datasets used to de-velop the MLPs contain no grain boundary structures.Therefore, the consistency indicates that the MLPs havehigh predictive power for the grain boundary structuresand their energies. The present results also imply thatthe MLPs in the repository, including those for othersystems, should be useful in accurately predicting grainboundary properties and other complex defect properties.
ACKNOWLEDGMENTS
This work was supported by a Grant-in-Aid for Sci-entific Research (B) (Grant Number 19H02419) and aGrant-in-Aid for Scientific Research on Innovative Areas(Grant Number 19H05787) from the Japan Society forthe Promotion of Science (JSPS). [1] A. P. Sutton and R. W. Balluffi,
Interfaces in crystallinematerials (Oxford: Clarendon Press, 1995).[2] J. Bishop and R. Hill, London, Edinburgh, Dublin Philos.Mag. J. Sci. , 414 (1951).[3] E. Kröner, Acta Metall. , 155 (1961).[4] Y. Mishin, M. Asta, and J. Li, Acta Mater. , 1117(2010).[5] J. R. Greer and J. T. De Hosson, Prog. Mater. Sci. ,654 (2011).[6] G. C. Hasson and C. Goux, Scr. Metall. , 889 (1971).[7] D. Wolf, Scr. Metall. , 1713 (1989).[8] K. L. Merkle and D. Wolf, MRS Bull. , 42 (1990).[9] M. W. Finnis and J. E. Sinclair, Philos. Mag. A , 45(1984). [10] M. S. Daw and M. I. Baskes, Phys. Rev. B , 6443(1984).[11] J. W. Cahn, Y. Mishin, and A. Suzuki, Philos. Mag. ,3965 (2006).[12] S. von Alfthan, K. Kaski, and A. P. Sutton, Phys. Rev.B , 134101 (2006).[13] J. A. Brown and Y. Mishin, Phys. Rev. B , 134118(2007).[14] M. Dao, L. Lu, R. J. Asaro, J. T. De Hosson, and E. Ma,Acta Mater. , 4041 (2007).[15] M. A. Tschopp and D. L. McDowell, Philos. Mag. ,3147 (2007).[16] D. L. Olmsted, S. M. Foiles, and E. A. Holm, Acta Mater. , 3694 (2009). (cid:104) (cid:105) STGB (cid:104) (cid:105)
STGB (cid:104) (cid:105) pure-twist grain boundaryAg Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Ag <100>STGBMLPAckland-1987Williams-2006DFT Σ Σ Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Ag <110>STGBMLPAckland-1987Williams-2006DFT Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Ag <100>pure-twistMLPAckland-1987Williams-2006DFT Al Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Al <100>STGBMLPMishin-1999DFT Σ Σ Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Al <110>STGBMLPMishin-1999DFT Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Al <100>pure-twistMLPMishin-1999DFT Au Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Au <100>STGBMLPAckland-1987Zhou-2004DFT Σ Σ Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Au <110>STGBMLPAckland-1987Zhou-2004DFT Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Au <100>pure-twistMLPAckland-1987Zhou-2004DFT Cu Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Cu <100>STGBMLPMishin-2001DFT Σ Σ Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Cu <110>STGBMLPMishin-2001DFT Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Cu <100>pure-twistMLPMishin-2001DFT Pd Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Pd <100>STGBMLPZhou-2004DFT Σ Σ Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Pd <110>STGBMLPZhou-2004DFT Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Pd <100>pure-twistMLPZhou-2004DFT Pt Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Pt <100>STGBMLPZhou-2004DFT Σ Σ Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Pt <110>STGBMLPZhou-2004DFT Σ G r a i n B ounda r y E ne r g y [ m J / m ] Rotation Angle [degree]Pt <100>pure-twistMLPZhou-2004DFT
Figure 2. Rotation angle dependence of the grain boundary energy for (cid:104) (cid:105)
STGBs, (cid:104) (cid:105)
STGBs, and (cid:104) (cid:105) pure-twist grainboundaries for elemental Ag, Al, Ag, Cu, Pd, and Pt, predicted using the MLPs. For comparison, the grain boundary energiespredicted using EAM potentials for Ag [57, 58], Al [59], Au [57, 60], Cu [61], Pd [60], and Pt [60] are shown by open symbols.The grain boundary energies computed by DFT calculation are also shown by crosses.
Table I. Model parameters of the MLPs used to estimate the grain boundary structures and energies. The identification of thefeature type, the model type, and the polynomial orders can be found in Ref. [47].Ag Al Au Cu Pd PtMLP-ID pair-44 gtinv-336 gtinv-111 pair-23 gtinv-722 gtinv-533RMSE (energy) [meV/atom] 2.2 0.8 0.7 2.2 6.3 12.9RMSE (force) [eV/Å] 0.010 0.008 0.012 0.013 0.097 0.172Time [ms/atom/step] [56] 0.05 1.85 0.66 0.04 0.52 0.63Number of coefficients 815 1100 475 285 500 1595Feature type Pair Invariants Invariants Pair Invariants InvariantsCutoff radius [Å] 7.0 8.0 6.0 7.0 6.0 6.0Number of radial functions 15 15 10 10 5 5Model type 2 3 3 2 4 2Polynomial order (function F ) 3 3 3 3 2 2Polynomial order (invariants) − − { l (2)max , l (3)max } − [4, 4] [4, 4] − [4, 0] [4, 2][17] M. D. Sangid, H. Sehitoglu, H. J. Maier, and T. Niendorf,Mater. Sci. Eng. A , 7115 (2010).[18] M. A. Tschopp, K. N. Solanki, F. Gao, X. Sun, M. A.Khaleel, and M. F. Horstemeyer, Phys. Rev. B , 064108(2012).[19] M. A. Tschopp, S. P. Coleman, and D. L. McDowell,Integr. Mater. Manuf. Innov. , 176 (2015).[20] S. Kiyohara, H. Oda, T. Miyata, and T. Mizoguchi, Sci.Adv. , e1600746 (2016).[21] G. S. Rohrer, E. A. Holm, A. D. Rollett, S. M. Foiles,J. Li, and D. L. Olmsted, Acta Mater. , 5063 (2010).[22] E. A. Holm, G. S. Rohrer, S. M. Foiles, A. D. Rollett,H. M. Miller, and D. L. Olmsted, Acta Mater. , 5250(2011).[23] P. Hohenberg and W. Kohn, Phys. Rev. , B864(1964).[24] W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965).[25] S. Lorenz, A. Groß, and M. Scheffler, Chem. Phys. Lett. , 210 (2004).[26] J. Behler and M. Parrinello, Phys. Rev. Lett. , 146401(2007).[27] A. P. Bartók, M. C. Payne, R. Kondor, and G. Csányi,Phys. Rev. Lett. , 136403 (2010).[28] J. Behler, J. Chem. Phys. , 074106 (2011).[29] J. Han, L. Zhang, R. Car, and W. E, Commun. Comput.Phys. , 629 (2018).[30] N. Artrith and A. Urban, Comput. Mater. Sci. , 135(2016).[31] N. Artrith, A. Urban, and G. Ceder, Phys. Rev. B ,014112 (2017).[32] W. J. Szlachta, A. P. Bartók, and G. Csányi, Phys. Rev.B , 104108 (2014).[33] A. P. Bartók, J. Kermode, N. Bernstein, and G. Csányi,Phys. Rev. X , 041048 (2018).[34] Z. Li, J. R. Kermode, and A. De Vita, Phys. Rev. Lett. , 096405 (2015).[35] A. Glielmo, P. Sollich, and A. De Vita, Phys. Rev. B ,214302 (2017).[36] A. Seko, A. Takahashi, and I. Tanaka, Phys. Rev. B ,024101 (2014).[37] A. Seko, A. Takahashi, and I. Tanaka, Phys. Rev. B ,054113 (2015).[38] A. Takahashi, A. Seko, and I. Tanaka, Phys. Rev. Mater. , 063801 (2017). [39] A. Thompson, L. Swiler, C. Trott, S. Foiles, andG. Tucker, J. Comput. Phys. , 316 (2015).[40] M. A. Wood and A. P. Thompson, J. Chem. Phys. ,241721 (2018).[41] C. Chen, Z. Deng, R. Tran, H. Tang, I.-H. Chu, and S. P.Ong, Phys. Rev. Mater. , 043603 (2017).[42] A. V. Shapeev, Multiscale Model. Simul. , 1153 (2016).[43] V. L. Deringer, C. J. Pickard, and G. Csányi, Phys. Rev.Lett. , 156001 (2018).[44] E. V. Podryabinkin, E. V. Tikhonov, A. V. Shapeev, andA. R. Oganov, Phys. Rev. B , 064114 (2019).[45] K. Gubaev, E. V. Podryabinkin, G. L. Hart, and A. V.Shapeev, Comput. Mater. Sci. , 148 (2019).[46] T. Mueller, A. Hernandez, and C. Wang, J. Chem. Phys. , 050902 (2020).[47] A. Seko, arXiv:2007.14206 (2020), Machine LearningPotential Repository at Kyoto University, https://sekocha.github.io/repository/index-e.html .[48] S. Plimpton, J. Comput. Phys. , 1 (1995).[49] S. P. Ong, W. D. Richards, A. Jain, G. Hautier,M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A.Persson, and G. Ceder, Comput. Mater. Sci. , 314(2013).[50] A. Seko, A. Togo, and I. Tanaka, Phys. Rev. B , 214108(2019).[51] P. E. Blöchl, Phys. Rev. B , 17953 (1994).[52] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[53] G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993).[54] G. Kresse and J. Furthmüller, Phys. Rev. B , 11169(1996).[55] G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999).[56] The computational time is estimated using a single coreof Intel Xeon E5-2695 v4 (2.10GHz).[57] G. J. Ackland, G. Tichy, V. Vitek, and M. W. Finnis,Philos. Mag. A , 735 (1987).[58] P. L. Williams, Y. Mishin, and J. C. Hamilton, Model.Simul. Mater. Sci. Eng. , 817 (2006).[59] Y. Mishin, D. Farkas, M. J. Mehl, and D. A. Papacon-stantopoulos, Phys. Rev. B , 3393 (1999).[60] X. W. Zhou, R. A. Johnson, and H. N. G. Wadley, Phys.Rev. B , 144113 (2004).[61] Y. Mishin, M. J. Mehl, D. A. Papaconstantopoulos, A. F.Voter, and J. D. Kress, Phys. Rev. B63