aa r X i v : . [ phy s i c s . c o m p - ph ] J u l Machine Learning Potential Repository
Atsuto Seko ∗ Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan (Dated: July 29, 2020)This paper introduces a machine learning potential repository that includes Pareto optimal ma-chine learning potentials. It also shows the systematic development of accurate and fast machinelearning potentials for a wide range of elemental systems. As a result, many Pareto optimal machinelearning potentials are available in the repository from a website [1]. Therefore, the repository willhelp many scientists to perform accurate and fast atomistic simulations.
I. INTRODUCTION
Machine learning potential (MLP) has been increas-ingly required to perform crystal structure optimizationsand large-scale atomistic simulations more accuratelythan with conventional interatomic potentials. There-fore, many recent studies have proposed a number ofprocedures to develop MLPs and have shown their ap-plications [2–23]. Simultaneously, MLPs themselves arenecessary for their users to perform accurate atomisticsimulations. Therefore, the development and distribu-tion of MLPs for a wide range of systems should be use-ful, similarly to the conventional interatomic potentialsdistributed in several repositories [24, 25].This study demonstrates an MLP repository availablefrom a website [1]. The MLP repository includes Paretooptimal MLPs with different trade-offs between accuracyand computational efficiency because they are conflictingproperties and there is no single optimal MLP [26–28].This study develops the repository by performing sys-tematic density functional theory (DFT) calculations forapproximately 460,000 structures and by combining themwith existing DFT datasets in the literature [26, 29].Polynomial-based potential energy models [26, 29] andtheir revisions are then systematically applied to the con-struction of MLPs for a wide range of elemental systems.Although the present version of the repository does notcontain MLPs for multicomponent systems, the reposi-tory will be gradually updated. Moreover, a user pack-age that combines MLPs in the repository with atomisticsimulations using the lammps code [30] is also availableon a website [31].
II. POTENTIAL ENERGY MODELS
This section shows structural features and potentialenergy models used for developing MLPs in the reposi-tory. Given cutoff radius r c from atom i in a structure,the short-range part of the total energy for the structure ∗ [email protected] may be decomposed as E = X i E ( i ) , (1)where E ( i ) denotes the contribution of atom i or theatomic energy. The atomic energy is then given by afunction of invariants for the O(3) group [26, 32] as E ( i ) = F (cid:16) d ( i )1 , d ( i )2 , · · · (cid:17) , (2)where d ( i ) n denotes an invariant derived from order pa-rameters representing the neighboring atomic density ofatom i . In the context of MLPs, invariants { d ( i ) n } canbe called “structural features”. Also, a number of func-tions are useful as function F to represent the relation-ship between the invariants and the atomic energy, suchas artificial neural network models [2, 3, 5–8], Gaussianprocess models [4, 9–12], and linear models [13–19]. Inthe repository, linear models are explained as function F ,which are shown in Sec. II B. A. Structural features
A systematic procedure to derive a set of structuralfeatures that can control the accuracy and computa-tional efficiency of MLPs (e.g., [26, 32]) plays an essen-tial role in automatically generating fast and accurateMLPs. Therefore, the repository employs systematic setsof structural features derived from order parameters rep-resenting the neighboring atomic density in terms of abasis set. They are classified into a set of structural fea-tures derived only from radial functions and a set of struc-tural features derived from radial and spherical harmonicfunctions.A pairwise structural feature is expressed as d ( i ) n = X j ∈ neighbor f n ( r ij ) , (3)where r ij denotes the distance between atoms i and j .The repository adopts a finite basis set of Gaussian-typeradial functions given by f n ( r ) = exp (cid:2) − β n ( r − r n ) (cid:3) f c ( r ) , (4)where β n and r n denote parameters. Cutoff function f c ensures smooth decay of the radial function, and therepository employs a cosine-based cutoff function ex-pressed as f c ( r ) = (cid:20) cos (cid:18) π rr c (cid:19) + 1 (cid:21) ( r ≤ r c )0 ( r > r c ) . (5) Another structural feature is a linearly independentpolynomial invariant of the O(3) group, which is gener-ated from order parameters representing the neighbor-ing atomic density in terms of spherical harmonics. A p th-order polynomial invariant for a given radial number n and a given set of angular numbers { l , l , · · · , l p } isdefined by a linear combination of products of p orderparameters, expressed as d ( i ) nl l ··· l p , ( σ ) = X { m ,m , ··· ,m p } C l l ··· l p , ( σ ) m m ··· m p a ( i ) nl m a ( i ) nl m · · · a ( i ) nl p m p , (6)where a ( i ) nlm denotes the order parameter of component nlm representing the neighboring atomic density of atom i . The order parameters for atom i in a given struc-ture are approximately calculated from its neighboringatomic density regardless of the orthonormality of theradial functions [26] as a ( i ) nlm = X j ∈ neighbor f n ( r ij ) Y ∗ lm ( θ ij , φ ij ) , (7)where ( r ij , θ ij , φ ij ) denotes the spherical coordinates ofneighboring atom j centered at the position of atom i .A coefficient set { C l l ··· l p , ( σ ) m m ··· m p } is determined by usinga group-theoretical projection operator method [33], en-suring that the linear combination of Eqn. (6) is invari-ant for arbitrary rotation [26]. In terms of fourth- andhigher-order polynomial invariants, there exist multipleinvariants that are linearly independent for most of theset { l , l , · · · , l p } . Therefore, they are distinguished byindex σ if necessary. B. Energy models with respect to structuralfeatures
The repository uses polynomial functions as function F representing the relationship between the atomic energyand a given set of structural features, D = { d , d , · · · } .The polynomial functions with regression coefficients { w } are given as follows. F ( D ) = X i w i d i F , pow ( D ) = X i w ii d i d i F ( D ) = X { i,j } w ij d i d j (8) F , pow ( D ) = X i w iii d i d i d i F ( D ) = X { i,j,k } w ijk d i d j d k ...A potential energy model is identified with a combina-tion of the polynomial functions and structural features.The repository introduces the following six potential en-ergy models. When a set of pairwise structural featuresis described as D ( i )pair = { d ( i ) n } , the first model ( modeltype = 1, feature type = pair ) is composed of pow-ers of the pairwise structural features as E ( i ) = F (cid:16) D ( i )pair (cid:17) + F , pow (cid:16) D ( i )pair (cid:17) + F , pow (cid:16) D ( i )pair (cid:17) + · · · , (9)which is measured from the energy of the isolated stateof atom i . This model was proposed in Refs. 13 and 14.The second model ( model type = 2, feature type =pair ) is a polynomial of the pairwise structural featureswith their cross terms, expressed as E ( i ) = F (cid:16) D ( i )pair (cid:17) + F (cid:16) D ( i )pair (cid:17) + F (cid:16) D ( i )pair (cid:17) + · · · . (10)This model can be regarded as a natural extension of em-bedded atom method (EAM) potentials as demonstratedin Ref. 15.The other four models are derived from the polynomialinvariants of Eqn. (6). When a set of the polynomialinvariants is expressed by the union of sets of p th-orderpolynomial invariants as D ( i ) = D ( i )pair ∪ D ( i )2 ∪ D ( i )3 ∪ D ( i )4 ∪ · · · , (11)where D ( i )2 = { d ( i ) nll } D ( i )3 = { d ( i ) nl l l } (12) D ( i )4 = { d ( i ) nl l l l , ( σ ) } , the third model ( model type = 1, feature type =invariants ) is expressed as E ( i ) = F (cid:16) D ( i ) (cid:17) + F , pow (cid:16) D ( i ) (cid:17) + F , pow (cid:16) D ( i ) (cid:17) + · · · . (13)This model consists of the powers of the polynomial in-variants. A linear polynomial form of the polynomialinvariants, E ( i ) = F (cid:0) D ( i ) (cid:1) , which was proposed in Ref.26, is included in the third model. Note that a linearpolynomial model with up to third-order invariants, ex-pressed by E ( i ) = F (cid:16) D ( i )pair ∪ D ( i )2 ∪ D ( i )3 (cid:17) , (14)is regarded as a spectral neighbor analysis potential(SNAP) [16].The fourth model ( model type = 2, feature type= invariants ) is given by a polynomial of the polyno-mial invariants as E ( i ) = F (cid:16) D ( i ) (cid:17) + F (cid:16) D ( i ) (cid:17) + F (cid:16) D ( i ) (cid:17) + · · · . (15)A quadratic polynomial model of the polynomial invari-ants up to the third order, expressed as E ( i ) = F (cid:16) D ( i )pair ∪ D ( i )2 ∪ D ( i )3 (cid:17) + F (cid:16) D ( i )pair ∪ D ( i )2 ∪ D ( i )3 (cid:17) , (16)is regarded as a quadratic SNAP [34].The fifth model ( model type = 3, feature type =invariants ) is the sum of a linear polynomial form ofthe polynomial invariants and a polynomial of pairwisestructural features, described as E ( i ) = f (cid:16) D ( i ) (cid:17) + f (cid:16) D ( i )pair (cid:17) + f (cid:16) D ( i )pair (cid:17) + · · · . (17)The sixth model ( model type = 4, feature type =invariants ) is the sum of a linear polynomial form ofthe polynomial invariants and a polynomial of pairwisestructural features and second-order polynomial invari-ants. This is written as E ( i ) = f (cid:16) D ( i ) (cid:17) + f (cid:16) D ( i )pair ∪ D ( i )2 (cid:17) + · · · . (18) III. DATASETS
Training and test datasets are generated from proto-type structures, i.e., structure generators. The reposi-tory uses two sets of structure generators for elementalsystems. One is composed of face-centered cubic (fcc), body-centered cubic (bcc), hexagonal close-packed (hcp),simple cubic (sc), ω , and β -tin structures, which was em-ployed in Ref. 29. Hereafter, structures generated fromthe structure generator set are denoted by “dataset 1”.The other is composed of prototype structures reportedin the Inorganic Crystal Structure Database (ICSD) [35],which aims to generate a wide variety of structures. Forelemental systems, only prototype structures composedof single elements with zero oxidation state are chosenfrom the ICSD. The total number of the structure gener-ators is 86. A list of structure generators can be found inthe Appendix of Ref. 26. Hereafter, structures generatedfrom the second set are denoted by “dataset 2”.Given a structure generator, the atomic positions andlattice constants of the structure generator are fully opti-mized by DFT calculation to obtain its equilibrium struc-ture. Then, a new structure is constructed by randomlattice expansion, random lattice distortion, and randomatomic displacements into a supercell of the structuregenerator. For a given parameter ε controlling the de-gree of lattice expansion, lattice distortion, and atomicdisplacements, the lattice vectors of the new structure A ′ and the fractional coordinates of an atom in the newstructure f ′ are given as A ′ = A + ε R (19) f ′ = f + ε A ′ − η , (20)where the (3 ×
3) matrix R and the three-dimensionalvector η are composed of uniform random numbers rang-ing from − A and vector f represent thelattice vectors of the original supercell and the fractionalcoordinates of the atom in the original supercell, respec-tively.For each elemental system, datasets 1 and 2 are com-posed of 3,000 and 10,000 structures, respectively, in ad-dition to the equilibrium structures of the structure gen-erators. Dataset 1 was developed in Ref. 29, whereasdataset 2, except for the case of elemental aluminum, isdeveloped in this study. Each of the datasets is then ran-domly divided into a training dataset and a test dataset.In the repository, datasets 1 and 2 are available for 31 and47 elements, respectively. This means that the repositorycontains MLPs developed from a total of 567,228 DFTcalculations.DFT calculations were performed using the plane-wave-basis projector augmented wave method [36] withinthe Perdew–Burke–Ernzerhof exchange-correlation func-tional [37] as implemented in the vasp code [38–40]. Thecutoff energy was set to 300 eV. The total energies con-verged to less than 10 − meV/supercell. The atomic po-sitions and lattice constants of the structure generatorswere optimized until the residual forces were less than10 − eV/˚A. IV. MODEL COEFFICIENT ESTIMATION
Coefficients of a potential energy model are estimatedfrom all the total energies, forces, and stress tensors in-cluded in a training dataset. Given a potential energymodel, therefore, the predictor matrix and observationvector are simply written in a submatrix form as X = X energy X force X stress , y = y energy y force y stress . (21)The predictor matrix X is divided into three submatri-ces, X energy , X force , and X stress , which contain structuralfeatures and their polynomial contributions to the totalenergies, the forces acting on atoms, and the stress ten-sors of structures in the training dataset, respectively.The observation vector y also has three components, y energy , y force , and y stress , which contain the total en-ergy, the forces acting on atoms, and the stress tensors ofstructures in the training dataset, respectively, obtainedfrom DFT calculations. Using the predictor matrix andthe observation vector, coefficients of a potential energymodel are estimated by linear ridge regression.In the case of dataset 2 for elemental aluminum, thetraining data has 9,086, 1,314,879, and 54,516 entriesfor the energy, the force, and the stress tensor, respec-tively. Therefore, the predictor matrix X has a size of(1 , , , n coeff ), where n coeff denotes the number ofcoefficients of the potential energy model and ranges from10 to 32 ,
850 in the potential energy models of the repos-itory.
V. PARETO OPTIMALITY
The accuracy and computational efficiency of thepresent MLP strongly depend on the given input parame-ters. They are (1) the cutoff radius, (2) the type of struc-tural features, (3) the type of potential energy model, (4)the number of radial functions, (5) the polynomial orderin the potential energy model, and (6) the truncationof the polynomial invariants, i.e, the maximum angularnumbers of spherical harmonics, { l (2)max , l (3)max , · · · , l ( p max )max } and the polynomial order of invariants, p max . Therefore,a systematic grid search is performed for each system tofind their optimal values. The input parameters used fordeveloping MLPs can be found in the repository.However, it is difficult to find the optimal set of pa-rameters because the accuracy and computational effi-ciency of an MLP are conflicting properties whose trade-off should be optimized, as pointed out in Ref. 26. Inthis multiobjective optimization problem involving sev-eral conflicting objectives, there is no single optimal so-lution but a set of alternatives with different trade-offsbetween the accuracy and the computational efficiency.In such a case, Pareto optimal points can be optimal solu-tions with different trade-offs [41]. Therefore, the repos- -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Al, Dataset 1)
MLP3 MLP2 MLP1 MLP1MLP2MLP3
TrainingTest
DFT (eV/atom) –4 –3 –2 –1 0 –4 –3 –2 –1 0–4 –3 –2 –1 0150–15 M L P – D FT ( m e V / a t o m ) FIG. 1. Distribution of MLPs in a grid search to find op-timal parameters controlling the accuracy and the computa-tional efficiency of the MLP for elemental Al. The elapsedtime for a single point calculation is estimated using a singlecore of Intel R (cid:13) Xeon R (cid:13) E5-2695 v4 (2.10 GHz). The red closedcircles show the Pareto optimal points of the distribution ob-tained using a non-dominated sorting algorithm. The cyanclosed circles indicate the MLP with the lowest prediction er-ror and two Pareto optimal MLPs with higher computationalcost performance. The distribution of the prediction errorsfor dataset 1 is also shown.TABLE I. Model parameters of MLP1, MLP2, and MLP3for elemental Al. MLP1 MLP2 MLP3Number of coefficients 2410 1770 815Feature type Invariants Pair PairCutoff radius 12.0 12.0 8.0Number of radial functions 20 20 15Model type 3 2 2Polynomial order (function F ) 3 3 3Polynomial order (invariants) 4 − −{ l (2)max , l (3)max , · · · } [4,4,2] − − itory contains all Pareto optimal MLPs for each systemand each dataset. -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Cu, Dataset 1) 0 5 10 15 20 2510 -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Mg, Dataset 1) 0 10 20 30 40 50 60 7010 -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Ti, Dataset 1) 0 10 20 30 40 50 60 7010 -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Y, Dataset 1) 0 5 10 15 20 2510 -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Ga, Dataset 1) 0 5 10 15 20 2510 -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Zn, Dataset 1) 0 10 20 30 40 50 60 7010 -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Nb, Dataset 1) 0 10 20 30 40 50 60 7010 -5 -4 -3 -2 P r ed i c t i on e rr o r ( m e V / a t o m ) Elapsed time (s/atom/MD step) (Single CPU core)Pareto optimal MLPs (Zr, Dataset 1)
MLP3 MLP2 MLP1MLP3 MLP2 MLP1MLP3 MLP2 MLP1 MLP3 MLP2 MLP1MLP3 MLP2 MLP1 MLP3 MLP2 MLP1MLP3 MLP2 MLP1 MLP3 MLP2 MLP1
FIG. 2. Distribution of MLPs in a grid search for elemental Cu, Ga, Mg, Zn, Ti, Zr, Y, and Nb. The closed red circles showthe Pareto optimal points of the distribution.
VI. MLPS IN REPOSITORY
Figure 1 shows the prediction error and the computa-tional efficiency of the Pareto optimal MLPs developedfrom dataset 1 for elemental Al. Figure 2 also showsthe Pareto optimal MLPs for elemental Cu, Ga, Mg, Zn,Ti, Zr, Y, and Nb. The prediction error is estimatedusing the root mean square (RMS) error of the energyfor the test dataset. The computational efficiency is es-timated using the elapsed time to compute the energy,the forces and the stress tensors of a structure with 284atoms. In Figs. 1 and 2, the elapsed time is normalizedby the number of atoms because it is proportional to thenumber of atoms as shown later. The behavior of therelationship between the prediction error and the com-putational efficiency for the other systems can be foundin the repository.Users of the repository can choose an appropriate MLPfrom the Pareto optimal ones according to their targetsand purposes. The MLP with the lowest prediction er-ror is denoted by “MLP1”, whereas two Pareto optimalMLPs showing higher computational cost performanceare denoted by “MLP2” and “MLP3”. As can be seenin Figs. 1 and 2, MLP2 and MLP3 exhibit high com-putational efficiency without significantly increasing theprediction error. This study introduces simple scores toevaluate the computational cost performance from theelapsed time t with the unit of ms/atom/step and theprediction error ∆ E with the unit of meV/atom. MLP2and MLP3 with higher computational cost performanceminimize t + ∆ E and 10 t + ∆ E , respectively.Figure 1 shows the distribution of the prediction errorsfor structures in dataset 1. Table I also lists the values ofthe model parameters of MLP1, MLP2, and MLP3. Thisinformation for the other Pareto optimal MLPs and theother systems can be found in the repository.Tables II and III list the prediction error and the com-putational efficiency of MLPs for each elemental systemobtained from datasets 1 and 2, respectively. MLP2 andMLP3 exhibit high computational efficiency while avoid-ing a significant increase of the prediction error. There-fore, MLP2 and MLP3 can be regarded as better poten-tials than MLP1 for most practical purposes.Figure 3 shows the elapsed times of single point calcu-lations for structures with up to 32,000 atoms using theEAM potential [42], MLP1, MLP2, and MLP3 for ele-mental Al. Structures were made by the expansion of thefcc conventional unit cell with a lattice constant of 4 ˚A.As can be seen in Fig. 3, linear scaling with respect to the number of atoms is achieved in all the MLPs. Althoughthe performance for only three MLPs is shown here, theother MLPs also exhibit linear scaling with respect tothe number of atoms. Therefore, the computational timerequired for a calculation of n step steps for a structurewith n atom atoms can be estimated as t × n atom × n step ,where t is the elapsed time per atom for a single pointcalculation listed in the repository. -6 -5 -4 -3 -2 -1 E l ap s ed t i m e ( s / M D s t ep ) Number of atoms
EAMMLP1 MLP3MLP2
FIG. 3. Dependence of the computational time requiredfor a single point calculation on the number of atoms. Theelapsed time is measured using a single core of Intel R (cid:13) Xeon R (cid:13) E5-2695 v4 (2.10 GHz).
VII. CONCLUSION
An MLP repository developed by a systematic applica-tion of the procedure to obtain Pareto optimal MLPs hasbeen demonstrated in this paper. In particular, MLPswith high computational cost performance, showing highcomputational efficiency without increasing the predic-tion error, are useful for most practical purposes. Cur-rently, many Pareto optimal MLPs are available in therepository from the website, and the number of MLP en-tries in the repository is continuously increasing. There-fore, the repository should be useful in performing accu-rate and fast atomistic simulations.
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. Seko, Machine Learning Poten-tial Repository at Kyoto University, https://sekocha.github.io/repository/index-e.html . [2] S. Lorenz, A. Groß, and M. Scheffler,Chem. Phys. Lett. , 210 (2004).[3] J. Behler and M. Parrinello, Phys. Rev. Lett. , 146401(2007). TABLE II. Prediction error and computational efficiency of MLPs constructed from dataset 1 for 31 elemental systems. Thenormalized elapsed time for a single point calculation, the RMS error for the energy, and the RMS error for the force aredenoted by t (s/atom/step), ∆ E (meV/atom), and ∆ f (eV/˚A), respectively. MLP1 shows the lowest prediction error of ∆ E .MLP2 and MLP3 show the lowest values of t + ∆ E and 10 t + ∆ E , respectively.MLP1 MLP2 MLP3 t ∆ E ∆ f t ∆ E ∆ f t ∆ E ∆ f Ag 10.71 1.9 0.004 0.07 2.0 0.008 0.03 2.2 0.011Al 6.74 0.5 0.006 0.30 0.9 0.014 0.07 1.8 0.016Au 21.65 0.5 0.006 0.66 0.7 0.012 0.05 1.8 0.027Ba 5.79 1.0 0.005 0.30 1.2 0.011 0.10 1.8 0.013Be 12.55 1.1 0.019 1.54 2.0 0.026 0.13 5.5 0.043Ca 15.83 1.0 0.004 0.07 1.1 0.011 0.07 1.1 0.011Cd 2.02 4.6 0.016 0.13 5.0 0.011 0.05 5.3 0.018Cr 12.64 2.8 0.061 2.31 3.6 0.070 0.80 5.5 0.082Cs 6.60 0.5 0.001 0.16 0.5 0.002 0.12 0.6 0.001Cu 6.82 1.7 0.004 0.10 2.0 0.011 0.03 2.2 0.013Ga 21.73 0.5 0.006 0.29 0.6 0.014 0.10 1.2 0.015Hf 21.72 0.9 0.039 1.85 1.4 0.051 0.18 4.3 0.103Hg 4.01 0.8 0.004 0.23 1.0 0.008 0.07 1.2 0.010In 21.67 0.5 0.005 0.22 0.7 0.014 0.07 1.2 0.014K 18.53 0.4 0.000 0.09 0.5 0.001 0.09 0.5 0.001Li 6.89 0.1 0.001 0.13 0.2 0.003 0.03 0.7 0.005Mg 21.71 0.4 0.002 0.18 0.5 0.006 0.10 0.6 0.007Mo 21.69 2.4 0.065 2.16 3.3 0.078 0.15 9.3 0.138Na 21.68 0.2 0.001 0.10 0.2 0.001 0.05 0.4 0.002Nb 21.65 2.4 0.048 2.18 2.8 0.058 0.10 9.0 0.127Rb 12.48 0.5 0.001 0.12 0.5 0.001 0.09 0.7 0.001Sc 21.71 0.7 0.017 0.22 2.6 0.048 0.18 3.0 0.049Sr 21.53 0.5 0.003 0.22 0.7 0.008 0.13 0.8 0.009Ta 21.70 1.6 0.056 0.91 3.3 0.071 0.22 8.5 0.126Ti 21.67 1.4 0.035 1.85 1.8 0.047 0.13 5.5 0.100Tl 21.66 0.5 0.005 0.29 0.8 0.012 0.10 1.6 0.014V 6.95 2.2 0.048 1.09 3.1 0.058 0.07 8.5 0.129W 21.69 2.9 0.080 2.32 3.9 0.092 0.14 12.0 0.177Y 12.77 0.8 0.016 0.29 2.2 0.040 0.13 3.3 0.044Zn 3.27 2.3 0.008 0.18 2.5 0.011 0.07 2.7 0.014Zr 6.71 1.4 0.044 1.84 2.4 0.055 0.05 11.0 0.116[4] A. 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Prediction error and computational efficiency of MLPs constructed from dataset 2 for 47 elemental systems.MLP1 MLP2 MLP3 t ∆ E ∆ f t ∆ E ∆ f t ∆ E ∆ f Ag 18.51 1.1 0.019 0.76 1.3 0.033 0.28 2.5 0.035Al 28.69 1.8 0.033 1.11 3.0 0.040 0.28 7.3 0.061As 23.39 5.1 0.125 1.98 8.5 0.144 0.58 13.5 0.178Au 23.49 3.1 0.028 0.75 3.9 0.035 0.28 7.0 0.056Ba 36.80 0.7 0.013 1.44 1.9 0.021 0.13 4.6 0.034Be 8.69 3.8 0.078 1.10 5.7 0.088 0.28 11.8 0.132Bi 23.45 2.8 0.130 2.06 4.6 0.121 0.62 9.4 0.166Ca 23.66 0.4 0.006 0.93 1.1 0.013 0.17 3.5 0.030Cd 23.41 0.7 0.011 0.75 1.8 0.018 0.28 3.2 0.026Cr 13.95 6.7 0.221 2.59 8.3 0.226 0.35 18.2 0.324Cs 37.17 0.4 0.001 0.22 0.6 0.002 0.09 0.9 0.003Cu 23.20 8.2 0.022 0.28 8.8 0.034 0.10 9.5 0.051Ga 23.38 1.1 0.028 1.12 1.8 0.039 0.27 4.5 0.044Ge 23.45 2.7 0.058 1.11 5.0 0.067 0.58 7.6 0.076Hf 13.65 4.2 0.121 2.51 6.0 0.137 0.75 8.6 0.148Hg 23.61 3.6 0.014 0.75 4.7 0.020 0.13 6.5 0.038In 23.59 0.7 0.016 0.93 1.2 0.021 0.28 2.8 0.028Ir 22.64 9.0 0.251 3.19 10.8 0.260 0.75 16.8 0.295K 7.10 0.1 0.001 0.22 0.4 0.002 0.08 0.7 0.003La 37.02 2.5 0.057 1.96 3.8 0.069 0.69 6.4 0.079Li 23.59 0.2 0.004 0.75 0.9 0.010 0.10 1.7 0.019Mg 23.62 0.3 0.006 0.75 0.8 0.009 0.14 2.9 0.029Mo 18.21 7.3 0.211 3.48 8.6 0.226 0.75 15.6 0.266Na 3.20 1.5 0.196 0.71 2.0 0.197 0.17 2.7 0.226Nb 36.63 6.5 0.182 2.55 7.6 0.183 0.75 11.8 0.212Os 14.15 10.2 0.304 3.21 11.4 0.300 0.75 18.8 0.348P 23.76 7.3 0.176 1.98 9.8 0.186 1.11 11.8 0.192Pb 23.98 1.2 0.028 0.94 2.4 0.037 0.17 5.0 0.057Pd 14.51 2.6 0.073 0.99 4.0 0.080 0.28 7.6 0.096Pt 23.19 5.3 0.137 1.11 7.3 0.148 0.72 8.2 0.156Rb 36.57 0.0 0.000 0.13 0.5 0.002 0.07 0.9 0.004Re 37.20 9.8 0.274 1.98 13.5 0.291 0.71 18.4 0.320Rh 13.78 6.4 0.186 1.98 8.5 0.192 0.71 12.6 0.217Ru 23.43 8.5 0.234 3.19 9.9 0.237 0.75 16.4 0.279Sb 23.50 3.4 0.120 2.00 6.0 0.411 0.75 8.7 0.124Sc 23.92 3.0 0.211 1.98 4.0 0.234 0.75 5.9 0.135Si 23.58 4.1 0.077 1.11 7.2 0.088 0.75 8.9 0.095Sn 23.45 1.7 0.036 1.12 3.5 0.049 0.58 5.5 0.061Sr 11.80 0.7 0.007 0.76 1.6 0.014 0.18 3.2 0.022Ta 22.94 6.5 0.190 3.17 7.7 0.195 0.75 12.3 0.221Ti 13.20 4.4 0.143 1.98 6.4 0.146 0.69 9.2 0.163Tl 24.14 0.8 0.015 0.72 2.2 0.023 0.15 5.0 0.060V 14.23 6.4 0.188 2.54 8.4 0.196 0.71 12.3 0.228W 22.71 8.3 0.247 3.17 9.8 0.254 0.99 14.7 0.286Y 36.55 2.6 0.050 1.98 3.9 0.062 0.71 6.7 0.070Zn 23.62 1.1 0.017 0.99 1.9 0.024 0.27 4.6 0.038Zr 14.57 5.9 0.130 0.82 9.0 0.139 0.75 9.1 0.140 [20] V. 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