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[Atsushi Togo](https://orcid.org/0000-0001-8393-9766), Atsuto Seko

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[On-the-fly training of polynomial machine learning potentials in computing lattice thermal conductivity](https://mdr.nims.go.jp/datasets/4f3f2aef-db8b-4247-b8ca-82a7740d6c6c)

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arXiv:2401.17531v3  [cond-mat.mtrl-sci]  13 May 2024On-the-fly training of polynomial machine learning potentials in computing latticethermal conductivityAtsushi Togo1, ∗ and Atsuto Seko21Center for Basic Research on Materials National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan2Department of Materials Science and Engineering,Kyoto University, Sakyo, Kyoto 606-8501, JapanThe application of first-principles calculations for predicting lattice thermal conductivity (LTC) incrystalline materials, in conjunction with the linearized phonon Boltzmann equation, has gained in-creasing popularity. In this calculation, the determination of force constants through first-principlescalculations is critical for accurate LTC predictions. For material exploration, performing first-principles LTC calculations in a high-throughput manner is now expected, although it requiressignificant computational resources. To reduce computational demands, we integrated polynomialmachine learning potentials on-the-fly during the first-principles LTC calculations. This paperpresents a systematic approach to first-principles LTC calculations. We designed and optimized anefficient workflow that integrates multiple modular software packages. We applied this approach tocalculate LTCs for 103 compounds of the wurtzite, zincblende, and rocksalt types to evaluate theperformance of the polynomial machine learning potentials in LTC calculations. We demonstrate asignificant reduction in the computational resources required for the LTC predictions.I. INTRODUCTIONCalculations of lattice thermal conductivity (LTC)based on first-principles calculations and the linearizedphonon Boltzmann equation [1–4] have become increas-ingly popular in recent years. This is because sufficientlyaccurate LTC values can be systematically predicted fora wide variety of crystals using available computer sim-ulation packages.[5–10] These computational tools areexpected to be applied in materials discovery withina high-throughput calculation environment. However,since first-principles LTC calculations are still compu-tationally intensive, there is a need for the developmentof methodologies to reduce the computational demands.We conventionally employ a supercell approach com-bined with the finite displacement method for first-principles LTC calculations. Random or systematic dis-placements are introduced to the supercells, and theforces on atoms are calculated using first-principles calcu-lations. Subsequently, supercell force constants are com-puted from the dataset composed of the displacementsand forces, and the LTC values are calculated from thesesupercell force constants. Many supercells with differentdisplacement configurations are often required to popu-late the tensor elements of the supercell force constants.The accuracy of predicting LTCs relies on the use offirst-principles calculations to obtain the displacement-force dataset. However, this approach is computation-ally intensive. In order to achieve precise LTC predic-tions with lower computational resources, compressivesensing force constants calculation methods were devel-oped, as reported in Refs. 11 and 12. These methods em-ploy regularized linear regression techniques to eliminatecertain tensor elements of the supercell force constants,∗ Author to whom any correspondence should be addressed. togo.atsushi@nims.go.jp.thereby reducing the required size of the displacement-force dataset.In this study, we introduce another approach to re-duce the computational demands of first-principles LTCcalculations. We incorporate polynomial machine learn-ing potentials (MLPs) [13, 14] into an intermediate stageof the LTC calculation process. The polynomial MLPsare trained using a small dataset of displacement-forcepairs and energies derived from first-principles calcula-tions. Subsequently, the polynomial MLPs generate alarge displacement-force dataset to calculate supercellforce constants with significantly lower computationaldemands than those required by first-principles calcula-tions. Compared to the compressive sensing approach,this approach may more efficiently represent energy sur-face of crystal potential. However, the workflow for per-forming the first-principles LTC calculation with this ap-proach is more complex. One objective of this studyis to encapsulate this intricate workflow into a softwarepackage, complete with a set of well-optimized defaultparameters. Developing this software package requires athorough understanding of the MLP code details. More-over, efficient estimations of polynomial MLPs can beaccomplished using linear regressions supported by pow-erful libraries for linear algebra. Therefore, we have uti-lized the polynomial MLP code developed by one of theauthors.The computational procedure is illustrated in Fig. 1.Initially, a set of supercells with random displacementsof atoms is prepared. Forces on atoms and energies inthe supercells are calculated using first-principles calcu-lations. The dataset, consisting of displacement-forcepairs and energies, is employed to train the polynomialMLPs. Forces on atoms in another set of supercells withrandom displacements of atoms are calculated using thetrained polynomial MLPs. Supercell force constants arethen calculated from the displacement-force dataset ob-tained through the polynomial MLPs. Finally, the LTChttp://arxiv.org/abs/2401.17531v3mailto:Author to whom any correspondence should be addressed. togo.atsushi@nims.go.jp.2values are calculated using the supercell force constantsobtained. Crystal symmetry plays an important role inreducing the computational demands and improving thenumerical accuracy.The goal of the methodological and software develop-ments presented in this study is to reduce the computa-tional demands of first-principles LTC calculations, withthe aim of high-throughput LTC calculations. At thesame time, we prioritize user convenience, consideringfactors such as calculation time and required memory.We have designed the workflow and software to make theuse of the polynomial MLPs appear straightforward fromthe user’s perspective. This study explores the feasibilityof employing polynomial MLPs as an intermediate stagein the calculation of third-order supercell force constantsfor first-principles LTC calculations.In this study, a systematic calculation of LTCs at 300K was performed for the same set of the 103 compoundsof wurtzite, zincblende, and rocksalt types reported inRef. 15. The computational workflow-design and detailsare presented in Secs. II and III. While most of the theo-retical and methodological background is covered in thereferenced articles, those relevant to this paper are de-scribed in Sec. III. The results of the LTC calculationsare summarized in Sec. IV.II. DESIGN OF COMPUTATIONALWORKFLOWAs shown in Fig. 1, our study utilized a specific ap-proach for LTC calculations. This approach consistsof specialized modules for each calculation step. Thesemodules are interconnected through a local file systemand data communication, including computer-networkfile transfers and application program interfaces (APIs).In the LTC calculation process, we generate two dis-tinct sets of supercells with random displacements ofatoms. This occurs in steps (a) and (d), as illustrated inFig. 1. In this study, we displaced atoms by a constantdistance in random directions. In step (b), we conductenergy and force calculations for the first set of supercellsusing first-principles calculations. This is the most com-putationally demanding step. The outputs of step (b)form a dataset used to train polynomial MLPs in step(c). In step (e), we calculate forces for the second set ofsupercells using the trained polynomial MLPs. In step(f), third-order supercell force constants are computedusing the displacement-force dataset from step (e). Fi-nally in step (g), LTC is calculated from the supercellforce constants. Use of crystal symmetry is important insteps (f) and (g) for the computational efficiency and nu-merical accuracy. The computational demand from step(c) to the end is negligible compared to step (b).Our computational workflow is specifically designedto optimize high-throughput LTC calculations, balancingefficiency and convenience. This workflow is divided intotwo main parts: dataset preparation and calculations us-ing this dataset. The dataset preparation stage involves aset of energy and force calculations using first-principlescalculations, which are normally distributed over com-puter nodes to conduct the calculations in parallel. Aftercompleting these calculations, we extract the necessarydata from the output files of the first-principles calcula-tions. This data is then saved for transfer via computer-network communication. In the subsequent steps, calcu-lations are performed on a single computer. Dependingon the operational requirements, data transfer betweenthe modules is facilitated either through APIs or via thelocal computer file system for ease of use.III. COMPUTATIONAL METHODSA. LTC calculationLTC values were computed by solving the Peierls-Boltzmann equation within the relaxation time approx-imation (RTA)[1, 2, 16] using the phono3py code.[9, 17]It is important to note that the method employed forthe LTC calculations may not be appropriate for com-pounds with low or high thermal conductivity amongthe 103 compounds selected. To simplify the method-ological investigation in this study, we considered onlyphonon-phonon scattering for determining the phononrelaxation times. Imaginary parts of phonon self-energiescorresponding to the bubble diagram were calculatedfrom supercell force constants, where a linear tetrahe-dron method was used to treat the delta function,[17]and their reciprocals were used as the relaxation times.Computational details on the calculation of the super-cell force constants are provided in the subsequent sec-tions. Phonon group velocities, mode heat capacities,frequencies, and eigenvectors were obtained from dy-namical matrices constructed using second-order super-cell force constants. Additionally, a non-analytical termcorrection [18–20] was applied to the dynamical matri-ces to account for long-range dipole-dipole interactionsin the harmonic phonon calculation. For the wurtzite-type compounds, reciprocal spaces were sampled using a19×19×10 mesh, while a 19×19×19 mesh was employedfor the zincblende-type and rocksalt-type compounds.B. Supercell force constants calculationForce constants required for predicting LTCs are de-termined as the coefficients Φlκα,... in the Taylor expan-sion of the potential energy V with respect to atomic3(a) Phonopy/Pypolymlp(f) Symfc(c) Pypolymlp(g) Phonopy & Phono3pySpglibDisplacements generationCrystal symmetry finding Force constants calculationMachine learning potential trainingForce calculationLattice thermal conductivity calculation(b) VASPEnergy and force calculation(e) PypolymlpSpglibCrystal symmetry findingAiiDA & AiiDA-VASPAutomationAPIAPIFileAPIFileAPIFile(d) Phonopy/PypolymlpDisplacements generationFile/APIFIG. 1. Schematic illustration of workflow employed in thisstudy for LTC calculations. Each calculation step is repre-sented by a box, within which the name of the software pack-age and the type of calculation are described. The arrowsroughly indicate flow of data. The data were passed via APIsor computer files. The steps are as follows: Step (a): Gen-eration of a modest number of supercells with random dis-placements. Step (b): Calculation of energies and forces inthe supercells generated in step (a) by first-principles calcu-lations. The submission of a large number of computationaljobs is automated using a workflow system. Step (c): Train-ing of polynomial MLPs using the energies and forces in thesupercells obtained in step (b). Step (d): Generation of alarge number of supercells with random displacements. Step(e): Calculation of forces in the supercells generated in step(d) using the trained polynomial MLPs. Step (f): Calcula-tion of force constants from the displacement-force datasetcalculated in step (e). Step (g): Calculation of LTC usingthe supercell force constants obtained in step (f). This studyintroduces steps (c), (d), and (e) to the workflow. Conven-tionally, the dataset from step (b) is used directly in step(f). This conventional approach is computationally demand-ing due to the necessity of a large dataset in step (b). Inthis workflow, second-order force constants can be calculatedin the conventional approach because the computational de-mand is much less than the calculation of third-order forceconstants in the conventional approach.displacements ulκα:V = Φ0 +∑lκαΦlκαulκα+12∑lκα,l′κ′α′Φlκα,l′κ′α′ulκαul′κ′α′+13!∑lκα,l′κ′α′,l′′κ′′α′′Φlκα,l′κ′α′,l′′κ′′α′′ulκαul′κ′α′ul′′κ′′α′′ + · · · , (1)where l, κ, and α represent the unit cell, atom indexwithin the unit cell, and Cartesian coordinate, respec-tively. By differentiating both sides of Eq. (1) with re-spect to ulκα, we obtain−flκα =Φlκα +∑l′κ′α′Φlκα,l′κ′α′ul′κ′α′+12∑l′κ′α′,l′′κ′′α′′Φlκα,l′κ′α′,l′′κ′′α′′ul′κ′α′ul′′κ′′α′′+ · · · , (2)where flκα represents the α-component of the force onatom lκ. In this study, we obtain these coefficients byfitting a dataset consisting of finite displacements andforces of atoms in supercells approximating Eq. (2). Theequation that we employ for fitting the third-order su-percell force constants is written as−Flκα =∑l′κ′α′ΦSClκα,l′κ′α′Ul′κ′α′+12∑l′κ′α′,l′′κ′′α′′ΦSClκα,l′κ′α′,l′′κ′′α′′Ul′κ′α′Ul′′κ′′α′′ ,(3)where Flκα and Ulκα represent the forces and displace-ments of atoms in supercells, respectively. ΦSClκα,l′κ′α′and ΦSClκα,l′κ′α′,l′′κ′′α′′ denote the second- and third-ordersupercell force constants, respectively. Here we assumeΦSClκα = 0.Higher-order terms are effectively included inΦSClκα,l′κ′α′ and ΦSClκα,l′κ′α′,l′′κ′′α′′ . In Eq. (3), the con-tributions from higher-order terms are expected tobecome negligible as the displacements become smaller.However, use of smaller displacements can make thecomputation of supercell force constants more suscepti-ble to numerical errors of forces. To find a compromisebetween these conflicting requirements, a modest inclu-sion of higher-order contributions is commonly adopted.Higher-order terms also introduce additional degreesof freedom. To average over them in the third-ordersupercell force constants, a larger displacement-forcedataset is required to achieve convergence in LTC values.The linear regression method was employed to calcu-late supercell force constants from the displacement-forcedataset. In this method, forces acting on atoms, whichwere randomly displaced from their equilibrium positions4in supercells, were computed either using the polynomialMLPs in our current approach or through first-principlescalculations in the conventional approach. Tensor ele-ments of supercell force constants were projected onto thesubspace defined by symmetry projection operators of to-tally symmetric irreducible representations of the spacegroup, index permutation, and translational invariance.In addition, detailed techniques were developed to en-hance the efficiency of this computational process. Thisprocess is implemented in the symfc code.[21, 22] Thecrystallographic symmetries were determined using thespglib code.[23]For the zincblende- and rocksalt-type compounds, weutilized supercells with 2×2×2 and 4×4×4 expansionsof the conventional unit cells to calculate third-order andsecond-order supercell force constants, respectively. Inthe case of the wurtzite-type compounds, supercells with3× 3× 2 and 5× 5× 3 expansions of the unit cells wereemployed for third-order and second-order supercell forceconstants calculations, respectively.To compute the second-order supercell force constants,we employed the finite difference method as imple-mented in the phonopy code.[15, 17] We used the samedisplacement-force datasets as those in Ref. 15, wherethe forces in these datasets had been computed throughfirst-principles calculations. The number of supercells inthe datasets were six, two, and two for the wurtzite-,zincblende-, and rocksalt-type compounds, respectively.Third-order supercell force constants were calculatedfrom the displacement-force datasets using the symfccode.[21] The numbers of symmetrically independentforce constant elements were 7752, 1536, and 758 forthe wurtzite-, zincblende-, and rocksalt-type compounds,respectively. These values were determined based onthe forces acting on atoms, which were inferred usingthe polynomial MLPs implemented in the pypolymlpcode.[14]C. Polynomial MLPsThe polynomial MLPs were trained using the datasetcomposed of forces and displacements of atoms and en-ergies in supercells. These energies and forces were com-puted through first-principles calculations. The perfor-mance of the polynomial MLPs in the LTC calculationvia the third-order supercell force constants is discussedin Section IV.For the 103 compounds, we trained the polynomialMLPs using the pypolymlp code.[14] In this training,Gaussian-type radial functions were employed, and thefunctional form fn(r) is given asfn(r) = exp[−βn(r − rn)2]fc(r), (4)fc(r) ={[cos(πr/rc) + 1]/2 (r ≤ rc),0 (r > rc).(5)where r represents the distance from the center of eachatom, and rc is the cutoff distance. βn and rn are theparameters, respectively. Radial functions with rc = 8.0Å and βn = 1.0 Å−2 and rn = (n − 1)(rc − 1.0)/11for n = 1, . . . , 12 were used. We considered polyno-mial invariants up to third order characterizing neigh-boring atomic density based on spherical harmonics withthe maximum angular numbers of spherical harmonicsl(2)max = l(3)max = 8. The polynomial models were thenconstructed by the polynomial functions of the pair in-variants and linear polynomial function of the polyno-mial invariants. We considered polynomial functions upto second order. The model coefficients were estimatedfrom electronic total energies and forces by the linearridge regression method.D. First-principles calculationFor the first-principles calculations, we employedthe plane-wave basis projector augmented wave (PAW)method [24] within the framework of DFT as imple-mented in the VASP code.[25–27] The generalized gradi-ent approximation (GGA) of Perdew, Burke, and Ernz-erhof revised for solids (PBEsol) [28] was used as the ex-change correlation potential. To ensure high numericalaccuracy in computing atomic forces, the projection op-erators were applied in reciprocal spaces and additionalsupport grids were employed for the evaluation of theaugmentation charges. Static dielectric constants andBorn effective charges were calculated with the conven-tional unit cells from density functional perturbation the-ory (DFPT) as implemented in the VASP code.[29, 30]A plane-wave energy cutoff of 520 eV was employedfor the supercell force calculations and 676 eV for theDFPT calculations. Reciprocal spaces of the zincblende-and rocksalt-type compounds were sampled by the half-shifted 2×2×2meshes for the 2×2×2 supercells, the half-shifted 1×1×1 meshes for the 4×4×4 supercells, and thehalf-shifted 8×8×8 meshes for the conventional unit cells.Reciprocal spaces of the wurtzite-type compounds weresampled by the 2×2×2 meshes that are half-shifted alongthe c∗ axis for the 3×3×2 supercells, the 1×1×2 meshesthat are half-shifted along the c∗ axis for the 5 × 5 × 3supercells, and the 12×12×8 meshes that are half-shiftedalong the c∗ axis for the unit cells.E. Automation of dataset preparationPerforming a large number of first-principles calcu-lations can be computationally intensive and may re-quire high-performance computing resources. This stageconsumes a significant amount of computational powerthroughout the LTC calculation process. It is virtuallyinevitable that some of these calculations fail for vari-ous reasons, such as reaching the maximum number ofelectronic structure convergence iterations or encounter-ing issues related to computer networks and hardware.5Although the proportion of failed calculations was rela-tively low, we have not yet fully automated error recoveryfor all possible cases.We systematically identified calculation failures and re-executed those calculations semi-manually with the as-sistance of the workflow system instead of attempting tofully automate all processes. After completing all the su-percell calculations using first-principles calculations, thedataset for each compound required for the subsequentLTC calculation process was composed into a single com-puter file in a structured format.For the systematic calculations of energies and forcesin supercells using first-principles calculations, we uti-lized the AiiDA environment [31–33] in conjunction withthe AiiDA-VASP plugin.[34] The advantage of using theworkflow automation system was not only the automa-tion of submitting calculation jobs to high-performancecomputers, but also the automated data storing of thecalculation results in a database systematically. Thecomputed data, stored within the AiiDA database, couldbe conveniently accessed through the Python program-ming language. By writing a concise Python script, wewere able to extract supercell energies, forces, and dis-placements from the AiiDA database on demand andconvert this data into the structured format required forimmediate use by the phono3py code.[9, 17] An exam-ple of the phono3py data format can be found in thephono3py github repository.F. Parameters for 103 binary compounds33 compounds for the wurtzite- and zincblende-typeand 37 compounds for the rocksalt-type were used toevaluate the LCT calculation approach proposed in thisstudy, and their chemical compositions are listed in Ta-bles I and II. Crystal structures of the wurtzite andzincblende types are similar, though their stacking or-ders are different, much like the relationship betweenface-centered-cubic and hexagonal-close-packed structuretypes. Since it is of interest to explore their similaritiesand differences in calculations, as also studied in Ref. 9,the compounds with the same chemical compositions forthe wurtzite and zincblende types were calculated. Thetables also provide information on lattice parameters, thechoices of PAW datasets from the VASP package, andelectronic total energies of the elements that were sub-tracted from the total energies of the compounds used totrain the polynomial MLPs.IV. RESULTSA. Choice of displacements and number ofsupercellsFor each compound, two distinct displacement-forcedatasets that share the same supercell basis vectors wereTABLE I. Lattice parameters, names of the VASP PAW-PBEdatasets, electronic total energies of the atoms used in thisstudy for 33 wurtzite- and zincblende-type compounds. w-a, w-c, and z-a denote the lattice parameters a and c of thewurtzite-type compounds, and a of the zincblende-type com-pounds, respectively.w-a w-c z-a energy (eV) energy (eV)AgI 4.56 7.45 6.44 (Ag pv) −0.233 (I) −0.182AlAs 4.00 6.58 5.67 (Al) −0.282 (As d) −0.989AlN 3.11 4.98 4.38 (Al) −0.282 (N) −1.905AlP 3.86 6.34 5.47 (Al) −0.282 (P) −1.140AlSb 4.35 7.16 6.17 (Al) −0.282 (Sb) −0.828BAs 3.35 5.55 4.77 (B) −0.359 (As d) −0.989BeO 2.70 4.38 3.80 (Be) −0.023 (O) −0.957BeS 3.41 5.63 4.84 (Be) −0.023 (S) −0.578BeSe 3.62 5.97 5.14 (Be) −0.023 (Se) −0.438BeTe 3.95 6.53 5.61 (Be) −0.023 (Te) −0.359BN 2.54 4.20 3.61 (B) −0.359 (N) −1.905BP 3.18 5.27 4.52 (B) −0.359 (P) −1.140CdS 4.13 6.72 5.84 (Cd) −0.021 (S) −0.578CdSe 4.31 7.03 6.09 (Cd) −0.021 (Se) −0.438CdTe 4.59 7.52 6.50 (Cd) −0.021 (Te) −0.359CuBr 3.92 6.48 5.56 (Cu pv) −0.274 (Br) −0.225CuCl 3.70 6.17 5.27 (Cu pv) −0.274 (Cl) −0.311CuH 2.81 4.44 3.93 (Cu pv) −0.274 (H) −0.946CuI 4.17 6.88 5.92 (Cu pv) −0.274 (I) −0.182GaAs 3.99 6.57 5.66 (Ga d) −0.286 (As d) −0.989GaN 3.18 5.18 4.50 (Ga d) −0.286 (N) −1.905GaP 3.83 6.31 5.44 (Ga d) −0.286 (P) −1.140GaSb 4.31 7.10 6.11 (Ga d) −0.286 (Sb) −0.828InAs 4.30 7.05 6.09 (In d) −0.264 (As d) −0.989InN 3.54 5.71 4.99 (In d) −0.264 (N) −1.905InP 4.15 6.81 5.88 (In d) −0.264 (P) −1.140InSb 4.60 7.56 6.52 (In d) −0.264 (Sb) −0.828MgTe 4.56 7.41 6.44 (Mg pv) −0.009 (Te) −0.359SiC 3.08 5.05 4.36 (Si) −0.522 (C) −1.340ZnO 3.24 5.23 4.56 (Zn) −0.016 (O) −0.957ZnS 3.79 6.21 5.36 (Zn) −0.016 (S) −0.578ZnSe 3.98 6.54 5.64 (Zn) −0.016 (Se) −0.438ZnTe 4.28 7.05 6.07 (Zn) −0.016 (Te) −0.359employed to calculate LTCs. Energies and forces of thesupercells in the first dataset were computed using first-principles calculations, while the polynomial MLPs wereutilized for calculating forces in the second dataset. Thefirst dataset was used to train the polynomial MLPs. Thesecond dataset was employed to compute third-order su-percell force constants by fitting.To investigate the performance of the polynomialMLPs in predicting LTC values, 100 supercells with ran-dom directional displacements were initially prepared asthe first dataset. Subsequently, the first 10, 20, 40, 60,6TABLE II. Lattice parameters a, names of the VASP PAW-PBE datasets, and electronic total energies of the atoms usedin this study for 37 rocksalt-type compounds.a energy (eV) energy (eV)AgBr 5.67 (Ag pv) −0.233 (Br) −0.225AgCl 5.44 (Ag pv) −0.233 (Cl) −0.311BaO 5.53 (Ba sv) −0.035 (O) −0.957BaS 6.36 (Ba sv) −0.035 (S) −0.578BaSe 6.58 (Ba sv) −0.035 (Se) −0.438BaTe 6.97 (Ba sv) −0.035 (Te) −0.359CaO 4.77 (Ca pv) −0.010 (O) −0.957CaS 5.63 (Ca pv) −0.010 (S) −0.578CaSe 5.87 (Ca pv) −0.010 (Se) −0.438CaTe 6.30 (Ca pv) −0.010 (Te) −0.359CdO 4.71 (Cd) −0.021 (O) −0.957CsF 5.96 (Cs sv) −0.166 (F) −0.556KBr 6.59 (K pv) −0.182 (Br) −0.225KCl 6.29 (K pv) −0.182 (Cl) −0.311KF 5.37 (K pv) −0.182 (F) −0.556KH 5.63 (K pv) −0.182 (H) −0.946KI 7.05 (K pv) −0.182 (I) −0.182LiBr 5.41 (Li sv) −0.286 (Br) −0.225LiCl 5.06 (Li sv) −0.286 (Cl) −0.311LiF 4.00 (Li sv) −0.286 (F) −0.556LiH 3.97 (Li sv) −0.286 (H) −0.946LiI 5.90 (Li sv) −0.286 (I) −0.182MgO 4.22 (Mg pv) −0.009 (O) −0.957NaBr 5.93 (Na pv) −0.246 (Br) −0.225NaCl 5.60 (Na pv) −0.246 (Cl) −0.311NaF 4.63 (Na pv) −0.246 (F) −0.556NaH 4.79 (Na pv) −0.246 (H) −0.946NaI 6.41 (Na pv) −0.246 (I) −0.182PbS 5.90 (Pb d) −0.374 (S) −0.578PbSe 6.10 (Pb d) −0.374 (Se) −0.438PbTe 6.44 (Pb d) −0.374 (Te) −0.359RbBr 6.88 (Rb pv) −0.168 (Br) −0.225RbCl 6.58 (Rb pv) −0.168 (Cl) −0.311RbF 5.66 (Rb pv) −0.168 (F) −0.556RbH 5.95 (Rb pv) −0.168 (H) −0.946RbI 7.32 (Rb pv) −0.168 (I) −0.182SrO 5.13 (Sr sv) −0.032 (O) −0.957and 80 supercells were selected from the list of 100 super-cells as subsets. Using the displacement-force pairs andenergies of these supercells, the polynomial MLPs weretrained, and the last 20 supercells were reserved as testdata to optimize their ridge regularization parameters.For the ease of use of the software package, we de-cided to employ a constant displacement distance, andto obtain reasonable LTC values, we chose a constantdisplacement distance of 0.03 Å. Interestingly, we foundthat the polynomial MLPs performed well even with arelatively large displacement distance, such as 0.1 Å. Itis important to note that these factors are highly de-pendent on the specific force calculators and calculationconfigurations used.We utilized another displacement-force dataset thatconsists of 400 supercells with random directional dis-placements for the computation of third-order supercellforce constants. These supercell forces were calculatedusing the trained polynomial MLPs, where the residualforces were subtracted. The root-mean-square errors ofthe polynomial MLPs trained on the 20 supercells rangedfrom approximately 5.5×10−6 to 1.4×10−3 eV/Å, whichare expected to represent the same degree of numericalerrors in the displacement-force dataset.Due to the numerical smoothness of the polynomialMLPs for the force calculation with respect to positionsof atoms compared to the first-principles calculations em-ployed in this study, we were able to choose a smallconstant displacement distance of 0.001 Å. This bene-fits better convergence with smaller dataset when fittingthe supercell force constants by Eq. (3). For instance,in the case of a displacement distance of 0.03 Å, it wasnecessary to employ 10000 supercells to achieve well con-verged LTC values for the 103 compounds. This suggeststhat when high-order force constants are more relevantfor specific compounds, direct calculation of third-ordersupercell force constants from the the displacement-forcedataset through first-principles calculations may requirea large dataset to achieve convergence of LTC values.B. Calculated LTCsIn Figs. 2, 3, and 4, we present the calculated LTCs ofthe 103 compounds at 300 K. We can see that datasetswith 20 supercells show good performance, at least forestimating LTC values roughly. In particular, the LTCvalues of most of the rocksalt-type compounds are wellrepresented by these small datasets. The wurtzite- andzincblende-type compounds exhibit similar tendencies inLTC values with respect to dataset size since these crystalstructures are similar. The datasets with 40 supercellsyield LTC results that are roughly converged.LTC values at 300 K predicted by the conventionalapproach, which directly uses the displacement-forcedataset obtained through first-principles calculations tofit third-order supercell force constants, are depicted bythe horizontal dotted lines. The third-order supercellforce constants were computed by the linear regressionmethod as implemented in the symfc code [21] fromthe first datasets with 100 supercells and 0.03 Å ran-dom directional displacements, which were those pre-pared for training the polynomial MLPs, as explained inSec. IVA. In addition, LTC values with 400 supercells forthe zincblende- and rocksalt-type compounds and thosewith 400 and 2000 supercells for the wurtzite-type com-pounds were also computed. These values are depicted7as horizontal lines in Figs. 2, 3, and 4. For most of thezincblende- and rocksalt-type compounds, LTC valuesderived from datasets with 100 supercells are found tobe adequate when compared to those from 400 super-cells. However, for the wurtzite-type compounds, evendatasets with 400 supercells are insufficient.The LTC values predicted for the wurtzite-type com-pounds using polynomial MLPs tend to align with thosecalculated directly from 2000 supercell datasets. Thisalignment emphasizes the utility and effectiveness of us-ing polynomial MLPs in these cases.C. Comparison with conventional LTC calculationIn Fig. 5, the LTC values of the 103 compounds calcu-lated through the polynomial MLPs trained using the20 supercell datasets are compared with those calcu-lated in the conventional approach using the same finite-difference displacement-force datasets [35] as those em-ployed in Ref. 15. These datasets share the same unitcells and supercell sizes for each compound. The latterdatasets for the wurtzite-, zincblende-, and rocksalt-typecompounds consist of 1254, 222, and 146 displacements,respectively, with a displacement distance of 0.03 Å.These displacements were systematically introduced con-sidering crystal symmetries [36] by using the phono3pycode.[9, 17] In all these calculations, the same versionof the phono3py code [9, 17] (release v3.0.3) was uti-lized to calculate the LTCs from the respective super-cell force constants. The results demonstrate that theLTC values obtained through the polynomial MLPs con-sistently agree with those predicted by the conventionalapproach.[15] The LTC values of the 103 compounds arealso tabulated in Tables III and IV.D. ConclusionTo improve the efficiency of high-throughput LTC cal-culations, we developed methodologies and modular soft-ware packages that utilize polynomial MLPs for com-puting LTC values based on first-principles calculation.We evaluated the feasibility of this computational ap-proach by calculating the LTCs of 103 compounds ofwurtzite, zincblende, and rocksalt types. This approachwas benchmarked against our previously used conven-tional approach. We found that this approach signifi-cantly reduces computational demands while maintaininga satisfactory accuracy level for LTC prediction. Apartfrom the initial stage of generating datasets using first-principles calculations, subsequent LTC calculation stepsrequire minimal computational resources. This enablesusers to calculate LTCs and various related physical val-ues on standard computers, given access to high-qualitydatasets. Our future plans include the computation anddistribution of such high-quality datasets for a wide rangeof compounds.TABLE III. LTC values of zincblende- and wurtzite-type com-pounds at 300 K in W/m·s as shown in Fig. 5 calculatedthrough the polynomial MLPs, z-κpMLP and w-κpMLP, re-spectively, and those by the conventional approach, z-κconvand z-κconv, respectively.z-κconv z-κpMLP w-κconvxxw-κconvzzw-κpMLPxxw-κpMLPzzAgI 3.538 3.260 2.491 3.008 2.229 2.654AlAs 79.84 79.88 68.18 66.24 62.99 61.21AlN 232.0 233.6 269.5 243.7 271.4 244.4AlP 79.18 79.21 71.84 65.52 71.17 65.54AlSb 91.03 91.19 52.59 49.54 49.09 46.89BAs 1614 1605 1535 1155 1541 1092BeO 361.7 361.0 317.8 325.0 316.8 322.9BeS 194.2 194.4 165.7 149.1 153.6 141.8BeSe 447.3 443.3 345.7 333.7 342.2 327.8BeTe 262.3 262.3 214.4 202.3 200.5 191.6BN 1301 1290 1058 1012 1043 972.2BP 489.5 488.3 498.5 359.9 468.0 338.1CdS 23.67 23.52 19.81 19.99 19.40 19.71CdSe 15.38 15.26 12.53 13.19 10.61 11.31CdTe 8.523 8.358 6.284 7.215 6.043 6.812CuBr 8.219 6.795 3.177 4.711 2.832 3.845CuCl 2.056 1.681 1.452 2.483 1.383 2.224CuH 21.44 21.16 6.248 5.724 6.202 5.949CuI 10.68 8.844 8.773 9.872 7.777 8.821GaAs 36.66 36.19 33.52 30.27 32.68 30.01GaN 263.1 266.7 318.0 308.3 306.9 289.8GaP 128.4 129.4 139.7 118.9 130.9 110.7GaSb 37.74 37.30 28.5 21.90 25.25 20.13InAs 25.47 25.35 24.23 23.28 22.92 22.08InN 103.3 103.7 116.8 120.6 112.1 113.2InP 83.65 85.97 75.73 72.38 70.61 63.27InSb 15.21 14.96 11.76 11.84 10.96 11.08MgTe 11.65 11.04 8.263 9.050 7.834 8.341SiC 459.9 460.8 512.6 426.4 508.4 425.1ZnO 64.27 63.47 51.38 59.91 50.80 57.65ZnS 60.17 58.97 59.81 63.11 55.63 59.30ZnSe 20.55 20.25 18.77 18.88 17.61 17.39ZnTe 29.79 31.08 21.54 21.43 19.78 19.56ACKNOWLEDGMENTSThis work was supported by JSPS KAKENHI GrantNumbers JP21K04632, JP22H01756, JP19H05787 and24K08021 Some of the calculations in this study were per-formed on the Numerical Materials Simulator at NIMS.8Number of supercells Number of supercells Number of supercellsNumber of supercells Number of supercells Number of supercells0 20 40 60 80 1000123LTC (W/m-K)w-AgI0 20 40 60 80 100020406080w-AlAs0 20 40 60 80 1000100200300w-AlN0 20 40 60 80 100020406080w-AlP0 20 40 60 80 1000204060w-AlSb0 20 40 60 80 100050010001500w-BAs0 20 40 60 80 1000100200300LTC (W/m-K)w-BeO0 20 40 60 80 100050100150w-BeS0 20 40 60 80 1000100200300400w-BeSe0 20 40 60 80 1000100200w-BeTe0 20 40 60 80 10005001000w-BN0 20 40 60 80 1000200400w-BP0 20 40 60 80 10005101520LTC (W/m-K)w-CdS0 20 40 60 80 100051015w-CdSe0 20 40 60 80 1000246w-CdTe0 20 40 60 80 10001234w-CuBr0 20 40 60 80 1000.00.51.01.52.0w-CuCl0 20 40 60 80 1000246w-CuH0 20 40 60 80 1000.02.55.07.510.0LTC (W/m-K)w-CuI0 20 40 60 80 1000102030w-GaAs0 20 40 60 80 1000100200300w-GaN0 20 40 60 80 100050100150w-GaP0 20 40 60 80 1000102030w-GaSb0 20 40 60 80 10001020w-InAs0 20 40 60 80 100050100LTC (W/m-K)w-InN0 20 40 60 80 100020406080w-InP0 20 40 60 80 1000510w-InSb0 20 40 60 80 1000.02.55.07.510.0w-MgTe0 20 40 60 80 1000200400w-SiC0 20 40 60 80 1000204060w-ZnO0 20 40 60 80 1000204060LTC (W/m-K)w-ZnS0 20 40 60 80 10005101520w-ZnSe0 20 40 60 80 10001020w-ZnTeFIG. 2. Filled circles show LTCs (κ) of the 33 wurtzite-type compounds calculated at 300 K with respect to the number ofsupercells in the datasets used to train the polynomial MLPs. The LTC values are the averages of the diagonal elements, i.e.,(2κxx + κzz)/3. The horizontal dotted, dashed-dotted, and dashed lines depict the LTC values calculated in the conventionalapproach from the datasets of 100, 400, and 2000 supercells without using the polynomial MLPs, respectively.[1] R. E. Peierls, Zur kinetischen theorie der wärmeleitungin kristallen, Ann. Phys. 395, 1055 (1929).[2] R. E. Peierls, Quantum theory of solids (Oxford Univer-sity Press, 2001).[3] J. M. Ziman, Electrons and phonons: the theory of transport phenomena(Oxford University Press, 1960).[4] G. P. Srivastava, Physics of phonons (CRC Press, 1990).https://doi.org/https://doi.org/10.1002/andp.19293950803https://doi.org/10.1093/acprof:oso/9780198507796.001.00019Number of supercells Number of supercells Number of supercellsNumber of supercells Number of supercells Number of supercells0 20 40 60 80 10001234LTC (W/m-K)z-AgI0 20 40 60 80 100020406080z-AlAs0 20 40 60 80 1000100200z-AlN0 20 40 60 80 100020406080z-AlP0 20 40 60 80 1000255075100z-AlSb0 20 40 60 80 100050010001500z-BAs0 20 40 60 80 1000100200300400LTC (W/m-K)z-BeO0 20 40 60 80 100050100150200z-BeS0 20 40 60 80 1000200400z-BeSe0 20 40 60 80 1000100200300z-BeTe0 20 40 60 80 100050010001500z-BN0 20 40 60 80 1000200400z-BP0 20 40 60 80 10001020LTC (W/m-K)z-CdS0 20 40 60 80 100051015z-CdSe0 20 40 60 80 1000.02.55.07.510.0z-CdTe0 20 40 60 80 1000.02.55.07.510.0z-CuBr0 20 40 60 80 100012z-CuCl0 20 40 60 80 10001020z-CuH0 20 40 60 80 1000.02.55.07.510.0LTC (W/m-K)z-CuI0 20 40 60 80 100010203040z-GaAs0 20 40 60 80 1000100200300z-GaN0 20 40 60 80 100050100150z-GaP0 20 40 60 80 100010203040z-GaSb0 20 40 60 80 1000102030z-InAs0 20 40 60 80 1000255075100LTC (W/m-K)z-InN0 20 40 60 80 1000255075100z-InP0 20 40 60 80 100051015z-InSb0 20 40 60 80 1000510z-MgTe0 20 40 60 80 1000200400z-SiC0 20 40 60 80 1000204060z-ZnO0 20 40 60 80 1000204060LTC (W/m-K)z-ZnS0 20 40 60 80 10005101520z-ZnSe0 20 40 60 80 1000102030z-ZnTeFIG. 3. Filled circles show LTCs of the 33 zincblende-type compounds calculated at 300 K with respect to the number ofsupercells in the datasets used to train the polynomial MLPs. The horizontal dotted and dashed-dotted lines depict the LTCvalues calculated in the conventional approach from the datasets of 100 and 400 supercells without using the polynomial MLPs,respectively.[5] T. Tadano, Y. Gohda, and S. Tsuneyuki, Anharmonicforce constants extracted from first-principles molecu-lar dynamics: applications to heat transfer simulations,J. Phys. Condens. Matter 26, 225402 (2014).[6] J. Carrete, B. Vermeersch, A. Katre, A. van Roekeghem,T. Wang, G. K. Madsen, and N. Mingo, almabte: A solver of the space-time dependent boltzmanntransport equation for phonons in structured materials,Comput. Phys. Commun. 220, 351 (2017).[7] A. Chernatynskiy and S. R. Phillpot,Phonon transport simulator (phonts),Comput. Phys. Commun. 192, 196 (2015).[8] O. Hellman, P. Steneteg, I. A. Abrikosov, andS. I. Simak, Temperature dependent effective potentialhttps://doi.org/10.1088/0953-8984/26/22/225402https://doi.org/https://doi.org/10.1016/j.cpc.2017.06.023https://doi.org/https://doi.org/10.1016/j.cpc.2015.01.00810Number of supercellsNumber of supercells Number of supercells Number of supercells Number of supercells Number of supercells0 20 40 60 80 1000.00.20.40.6LTC (W/m-K)r-AgBr0 20 40 60 80 1000.00.20.40.6r-AgCl0 20 40 60 80 1000123r-BaO0 20 40 60 80 10002468r-BaS0 20 40 60 80 1000510r-BaSe0 20 40 60 80 1000510r-BaTe0 20 40 60 80 1000102030LTC (W/m-K)r-CaO0 20 40 60 80 100010203040r-CaS0 20 40 60 80 10005101520r-CaSe0 20 40 60 80 1000510r-CaTe0 20 40 60 80 1000.02.55.07.510.0r-CdO0 20 40 60 80 1000.00.51.01.52.0r-CsF0 20 40 60 80 1000123LTC (W/m-K)r-KBr0 20 40 60 80 10002468r-KCl0 20 40 60 80 10002468r-KF0 20 40 60 80 1000510r-KH0 20 40 60 80 100012r-KI0 20 40 60 80 1000123r-LiBr0 20 40 60 80 1000246LTC (W/m-K)r-LiCl0 20 40 60 80 10005101520r-LiF0 20 40 60 80 100010203040r-LiH0 20 40 60 80 100012r-LiI0 20 40 60 80 1000204060r-MgO0 20 40 60 80 1000123r-NaBr0 20 40 60 80 10002468LTC (W/m-K)r-NaCl0 20 40 60 80 10001020r-NaF0 20 40 60 80 10005101520r-NaH0 20 40 60 80 100012r-NaI0 20 40 60 80 100012r-PbS0 20 40 60 80 100012r-PbSe0 20 40 60 80 1000123LTC (W/m-K)r-PbTe0 20 40 60 80 10001234r-RbBr0 20 40 60 80 1000123r-RbCl0 20 40 60 80 10001234r-RbF0 20 40 60 80 1000246r-RbH0 20 40 60 80 1000.00.51.01.52.0r-RbI0 20 40 60 80 1000510LTC (W/m-K)r-SrOFIG. 4. Filled circles show LTCs of the 37 rocksalt-type compounds calculated at 300 K with respect to the number ofsupercells in the datasets used to train the polynomial MLPs. The horizontal dotted and dashed-dotted lines depict the LTCvalues calculated in the conventional approach from the datasets of 100 and 400 supercells without using the polynomial MLPs,respectively.11 1 10 100 1000 1  10  100  1000AgBr AgClBaOBaSBaSeBaTeCaOCaSCaSeCaTeCdOCsFKBrKCl KFKHKILiBrLiClLiFLiHLiIMgONaBrNaClNaFNaHNaIPbSPbSePbTeRbBrRbClRbFRbHRbISrOAgIAlAsAlNAlPAlSbBAsBeOBeSBeSeBeTeBNBPCdSCdSeCdTeCuBrCuClCuHCuIGaAsGaNGaPGaSbInAsInNInPInSbMgTeSiCZnOZnSZnSeZnTeAgIAlAsAlNAlPAlSbBAsBeOBeSBeSeBeTeBNBPCdSCdSeCdTeCuBrCuClCuHCuIGaAsGaNGaPGaSbInAsInNInPInSbMgTeSiCZnOZnSZnSeZnTeLTC calculated by conventional approach (W/m∙s)LTC calculated through polynomial MLPs (W/m∙s)Wurtzite typeZincblende typeRocksalt typeFIG. 5. 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