Developing fast and accurate methods to discover intermetallic compounds is relevant for alloy design. While density-functional-theory (DFT)-based methods have accelerated design of binary and ternary alloys by providing rapid access to the energy and properties of the stable intermetallics, they are not amenable for rapidly screening the vast combinatorial space of multi-principal element alloys (MPEAs). Here, a machine-learning model is presented for predicting the formation enthalpy of binary intermetallics and is used to identify new ones. The model uses easily accessible elemental properties as descriptors and has a mean absolute error of 0.025 eV/atom in predicting the formation enthalpy of stable binary intermetallics reported in the Materials Project database. The model further predicts stable intermetallics to form in 112 binary alloy systems that do not have any stable intermetallics reported in the Materials Project database. DFT calculations confirm one such stable intermetallic identified by the model, NbV_{2}, to be on the convex hull. Furthermore, an adaptive transfer learning method is used to generalize the model to predict ternary intermetallics with a similar accuracy as DFT, which suggests that it could be extended to identify compositionally complex intermetallics that may form in MPEAs.

## INTRODUCTION

Intermetallics are metallic alloys that have a fixed composition and an ordered crystal structure. They form a diverse class of compounds with over 20 000 known compositions crystallizing in over 2100 structure types, with new ones being discovered constantly.^{1} The presence of long range order and mixed bonding (metallic with ionic or covalent) in intermetallics distinguishes them from conventional metallic alloys in terms of their physical and mechanical properties. A common example is Ni_{3}Al that crystallizes into an ordered $L12$ structure (according to the Pearson notation)^{2} with Al atoms occupying the corners of a cube and Ni atoms occupying the face centers. As precipitates, it strengthens nickel-based superalloys for high-temperature applications.^{3,4} In contrast, the formation of brittle intermetallics in the Au–Al systems, such as AuAl_{2} (cF12-CaF_{2} prototype), is a significant cause of wire bonding failures in microelectronics.^{5,6} Besides their mechanical properties,^{7,8} intermetallics are widely studied as shape memory alloys,^{9} superconductors,^{10} and catalysts.^{11,12} With the rapid emergence of multi-principal element alloys (MPEAs) and especially high-entropy alloys (HEAs)^{13}—that form a single-phase solid solution on mixing five or more elements at a high (near equiatomic) concentration—knowledge of the intermetallics that can form in such compositionally complex systems is vital to predict their microstructures and properties. For instance, the strengthening of ductile fcc CoCrFeNiMo* _{x}* MPEAs has been attributed to the precipitation of hard

*σ*and

*μ*intermetallic phases, without concomitant embrittlement.

^{14}Meanwhile, Troparevsky

*et al*. have shown that the tendency of an MPEA to form a single-phase solid solution (HEA) can be predicted based on the formation enthalpy (

*ΔH*) of pairwise binary intermetallics.

_{f}^{15}Therefore, fast and accurate prediction of intermetallics and their

*ΔH*is of practical interest. But, it is challenging given the vast combinatorial space involving 81 elements and over 2100 structure types.

_{f}^{1}

To efficiently navigate through this expansive chemical and structural space and to rapdily screen intermetallic compounds, several strategies have been adapted. These range from empirical rules to high-throughput total energy calculations to machine learning (ML) models applied to materials databases.^{16–19} Empirical, valence electron-counting rules have been successful in identifying elemental combinations that may form stable intermetallics within specific families such as Zintl and Heusler phases.^{20,21} With increasing computing power, first-principles density-functional-theory (DFT)-based high-throughput total energy calculations have allowed successful identification of new intermetallics without being confined to any particular structure type; however, they are inefficient to search for ternary and more chemically complex intermetallics where the configurational space and the computational expense explodes.^{22–24} The availability of open-source materials databases, often built on results obtained from high-throughput DFT calculations, such as Materials Project (http://materialsproject.org/),^{22} NOMAD (http://nomad-coe.eu/),^{25} and OQMD (http://oqmd.org/),^{23} have enabled the use of data-centric informatics methods—popularly called materials informatics—to identify new materials or predict unknown properties.^{26} There are three ways to search for intermetallics using materials informatics: screening based on structure propotypes,^{27} screening based on chemical compositions,^{18} or a combination of the two.^{19} While predicting new intermetallics with structure propotypes has efficiently reduced the workload in interested systems,^{19,27} it is limited to common structure types. Here, our goal is to develop a machine learning model that can realize fast screening of intermetallics with good accuracy, given just chemical composition.

In this article, we present a ML model to accelerate the discovery of intermetallics by predicting their *ΔH _{f}* based on composition. The model was trained on DFT-calculated

*ΔH*of stable intermetallics available in the Materials Project, which lists the change in enthalpy upon forming an intermetallic with respect to the enthalpy of the constituent elements in their standard states.

_{f}^{22}We described metallic compositions using easily accessible elemental properties through the Matminer package

^{28}and found properties like valence electron concentration, electronegativity, cohesive energy, and first ionization energy to be the primary features of the model. The model can predict $\Delta Hf$ for binary intermetallics with a mean absolute error (MAE) of 0.022 eV/atom for the training set and an MAE of 0.044 meV/atom for the testing set. The model further predicts new, stable intermetallics to form in 112 metallic pairs, where intermetallics have not been reported, without providing information about their crystal structures. We confirmed the stability of one of the predicted new binary intermetallics NbV

_{2}using DFT calculations and found it to exist as stable Laves phases. We applied the binary model to predict ternary intermetallics and found that it reproduces Δ

*H*of known compounds with a mean absolute error (MAE) of 0.085 eV/atom and 0.057 eV/atom, without and with further training, respectively. Based on this result, we posit that this model can be extended to guide the prediction of compositionally complex intermetallics that may form in MPEAs.

_{f}## METHODS

We developed the model using the following three steps: building a dataset of binary intermetallics, describing the composition of each intermetallic with numerical attributes based on elemental properties, and mapping the attrubutes to *ΔH _{f}* through an ML algorithm. The workflow for the development of the model is shown schematically in Fig. 1. We provide details about these three steps in the following.

### Datasets

We picked binary intermetallics comprised of 48 metallic elements, including alkali metals, alkaline-earth metals, transition metals, post-transition metals, lanthanum, and actinium. These elements have been shaded with color in the periodic table shown in Fig. 1(a). We queried the Materials Project database for DFT-calculated *ΔH _{f}* of all the stable binary intermetallics

^{22}using the Materials Application Programming Interface.

^{29}Figure 1(b) shows a typical binary phase diagram, in this case of Ti–Al, obtained from the Materials Project. The solid black line indicates the convex hull that connects all the stable phases. Phases above the convex hull are metastable and are expected to decompose into adjacent stable phases under thermodynamic equillibrium. While training our model, we only include 1538 stable binary intermetallics that are on the convex hull. The frequency with which the 48 elements form stable binary intermetallics is shown in Fig. 1(a) using a colormap. In a similar manner, we queried available phase diagrams of ternary systems in the Materials Project and imported a list of 2118 stable ternary intermetallics for evaluation.

### Compositional representation

To develop an ML model, it is necessary to represent the various intermetallics numerically using one or more quantitative attributes, which are also called descriptors or features. The choice of descriptors is highly dependent on the property to be modeled. As we are interested in predicting the formability of intermetallics for any given combination of elements, we have used elemental properties to transform the chemical composition of intermetallics into numerical descriptors. We start by indexing several elemental properties, including Zunger's pseudopotential radius R_{s + p} (R_{s + p}),^{30} electronegativity (EN), molar volume (MV), melting temperature (MT), 1^{st} ionization energy (FIE), number of valence electrons (VEC), the heat of vaporization (HV), electron affinity (EA), atomic radius (AR) as saved in the Magpie preset in Matminer,^{28,31} and bulk modulus (BM) and cohesive energy (CE) collected from the literature.^{32} We used the Matminer package to transform the elemental properties to compositional descriptors. For every alloy composition, we map each elemental property using four attributes: the mean, mean absolute deviation, maximum, and minimum value of the composition's consitituent elements, which results in a total of 44 descriptors. To avoid linearly correlated descriptors, we calculated the Pearson correlation between each pair of descriptors. A correlation map for the mean attributes of the elemental properties is shown in Fig. 1(c). The colormap indicates the Pearson correlation coefficient. This coefficient ranges from −1 to 1, with −1 denoting the total negative linear correlation, 0 representing no linear correlation, and 1 showing the total positive linear correlation. For those elemental properties that are very strongly correlated (Pearson correlation coefficient larger than 0.9),^{33} one set is eliminated based on domain knowledge. For example, cohesive energies of elements show a strong linear relatonship with their heats of vaporization, as shown in Fig. 1(d). We retain only cohesive energy as a feature since it represents the energy gained by arranging atoms to a crystalline state, while heat of vaporization is the energy needed to transfer a liquid to its gaseous state. After removing highly correlated elemental properties, we are left with 32 descriptors. None of these elemental properties show a strong linear correlation with $\Delta Hf$; as an example, we show the variation of *ΔH _{f}* with the mean pseudopotential radius R

_{s + p}in Fig. 1(e). Therefore, we use nonlinear regression functions as discussed below.

### Statistical learning methods

We employed gaussian process regression (GPR) as implemented in the Scikit-learn python package to learn the nonlinear relationship between an intermetallic's descriptors and its *ΔH _{f}*.

^{34}For a given set of numerical descriptors, GPR learns a multivariate gaussian distribution that maps them to the target property (

*ΔH*). We use a sum-kernel which consists of a squared-exponential kernel and a white kernel that represents noise,

_{f}where $xiandxj$ are descriptor vectors for two intermetallics $iandj$, respectively, and signal variance $\sigma $, length-scale parameter *l*, and noise level $\sigma n$ are hyperparameters to be determined during the training process. We also tested other nonlinear regression methods with the same kernel, such as kernel ridge regression.^{35} We chose GPR because it learns a generative, probabilistic model of the target function and can thus provide meaningful uncertainties/confidence intervals along with the predictions.^{36} Also, GPR can choose the kernel's hyperparameters based on gradient-ascent on the marginal likelihood function, which is relatively fast compared to similar models.

Before training the model, all descriptors are rescaled to a range [0, 1] with min–max normalization. Here, elements $xi$ of a descriptor vector *x* = (*x*_{1}, *x*_{2}, …, *x*_{n}) are normalized to $xi\u2032$ with $xi\u2032=xi\u2212min(x)max(x)\u2212min(x)$. We used 48 metallic elements and 80% of the binary dataset for model training (training set) and kept the remaining 20% for model evaluation (testing set). To generalize the model and estimate model performance on new data, we applied 10-fold cross-validation (CV) during the training process, which randomly splits the data into 10 subsets and iteratively fits the model with nine of them and evaluates on the remaining subset. The accuracies are averaged for the entire process and defined as CV accuracy. This process is shown schematically in Fig. 1(f).

To establish the feasibility of extending the GPR model based on binary/ternary intermetallics to compositionally complex intermetallics where limited data are available, we have used an adaptive transfer learning algorithm based on Gaussian processes (AT-GP), as implemented in the GPflow package.^{37,38} Considering an established GPR model with a large amount of training data as the source task, and a similar regression problem with a small amount of training data as the target task, AT-GP automatically measures the similarity between the two tasks and learns a new model based on the conditional distribution of the target task, given the source information.

### Computational methods

We performed the DFT calculations using the Vienna *Ab initio* Simulation Package (VASP) with projector augmented-wave potentials.^{39,40} For the search of stable intermetallic structures, we employed the generalized gradient approximation (GGA) as implemented in the Perdew−Burke−Ernzerhof (PBE) functional.^{41} For *ΔH _{f}* calculations, we used a plane-wave basis set with a cut-off energy of 400 eV and performed relaxation until the Hellmann–Feynman forces on the atoms are less than 0.001 eV/Ǻ. The Brillouin zone was sampled using a Monkhorst−Pack

*k*-points mesh, while keeping the number of

*k*-points times lattice constant equal to ∼30 and ∼80 for structural relaxation and the single-step static calculation, respectively.

^{42}The phonon calculations were performed using the frozen-phonon approach, and the dispersion spectra were calculated using the Phonopy package.

^{43}For accurate phonon calculations, a higher cut-off energy of 700 eV for the plane-wave basis set was used with a tighter electronic convergence of $10\u22128eV$. Additionally, to calculate the force-constant matrices, a 2 × 2 × 2 supercell was used for the C15 phase and a 3 × 3 × 2 supercell was used for the C14 phase.

## RESULTS AND DISCUSSION

To identify the most important set of descriptors that can accurately predict *ΔH _{f}*, we employed forward and backward feature selection during the model development. As mentioned above, we use four attributes (mean, mean absolute deviation, maximum, and minimum) of each elemental property as descriptors to distinguish between different compositions of any pair of elements. In the forward selection, we start with an empty feature set and iterate through the various features and select one that maximizes the CV accuracy. We found that the VEC, i.e., the mean, mean absolute deviation, maximum, and minimum VEC values, leads to the highest CV accuracy of 0.64, as measured by the coefficient of determination $R2$. We then kept these four attributes in the feature set and searched for a second set of attributes of an elemental property that maximizes the CV accuracy. We cycled this process until the addition of a new property did not further improve the performance of the model. This process is shown in Fig. 2(a) with the colormap indicating CV accuracy and yellow stars indicating the elemental property that is selected in each iteration. We kept those elemental properties that are selected in previous iterations in the model and have labeled them with blue stars. Using this approach, we reduced the total number of descriptors from 32 to 16, with the best subset including VEC, FIE, EN, and CE. The elemental properties are indexed in the table in Fig. 2(a). The improvement in the accuracy of the GPR model with an increasing number of descriptors (also the number of iterations) during the forward feature selection process is shown in Fig. 2(b). The coefficient of determination $R2$ for the 10-fold CV (average accuracy for the 10 CV iterations) prediction on training and testing data increases until the model reaches 16 decriptors, after which it either plateaus or decreases slightly due to overfitting.

To further validate the importance of the descriptors down-selected from forward selection and ascertain that important descriptors were not left out, we used backward eliminaton. Here, we start with all the 32 descriptors and iteratively drop the one that has the least effect on accuracy, until a significant drop is observed on further removal of any descriptor. The variation of accuracies during this process is shown in Fig. 2(c), which also reduced to 16 descriptors with the same subset as forward selection.

In the above selection process, we simultaneously added or eliminated all four attributes of any elemental property. To further reduce the risk of overfitting, we then performed feature selection with respect to individual attributes of the down-selected elemental properties. We found that the mean absolute deviation attributes for elemental properties did not play a role in improving the model performance, which reduces the number of descriptors to 12. With the mean, maximum, and minimum attributes of VEC, FIE, EN, and CE as final features, we observe model performance benefits by adding more training data, as shown in Appendix Fig. 6(a). With 80% of the binary intermetallics as training samples, we obtained a mean absolute error (MAE) of 0.022 ± 0.001 eV/atom for the training set and a MAE of 0.044 ± 0.005 eV/atom for the testing set. For our final 12-feature model, we show the predicted Δ*H _{f}* with respect to the DFT-calculated value for training and testing sets, separately, in Fig. 2(d). For reference, the MAE of the DFT calculated

*ΔH*with respect to experimental measurements is ∼0.145 eV/atom for entries in the Materials Project database, when using the elemental DFT total energies as chemical potentials.

_{f}^{44,45}Furthermore, 80% of the binary intermetallics are predicted within an absolute error of 0.025 eV/atom using our model for the testing set, as shown in Fig. 2(e). These results indicate that the model predictions are reliable with a DFT-level accuracy and it can potentially be applied for the discovery of new intermetallic compositions.

To further establish the generality of our model to predict *ΔH _{f}* of complex intermetallics, we evaluate the model that was trained with binary intermetallics with a list of 2118 stable ternary intermetallics obtained from the Materials Project. The 12-feature model, without any further training, gives a MAE of 0.085 eV/atom in the predicted Δ

*H*of the ternary intermetallics. Figure 2(f) shows the histogram of the absolute error compared to DFT-calculated

_{f}*ΔH*for ternary intermetallics.

_{f}We then attempt to improve the model performance on ternary intermetallics by adding additional descriptors or including a fraction of ternary intermetallics as a training sample. During the feature selection process, we noticed that although the CV accuracy for the model to predict Δ*H _{f}* of binary intermetallics saturates with ∼12 descriptors, its prediction accuracy in the case of ternary intermetallics increases with more descriptors, as shown in the forward feature selection curve in Fig. 3(a). By increasing the number of descriptors from 12 to 24, MAE in Δ

*H*of ternary intermetallics decreases from 0.085 eV/atom to 0.057 eV/atom. The additional descriptors are the mean, maximum, and minimum attributes of molar volume (MV), bulk modulus (BM), melting temperature (MT), and atomic radius (AR).

_{f}Meanwhile, we keep the binary intermetallics in the training set and test the model performance by adding varying amount of ternary intermetallics as training samples. We use the remaining ternary intermetallics as the validation set. We observe that the validation accuracy of the 24-feature model improves with increasing amount of ternary training samples until it reaches the highest accuracy with 80% ternary intermetallics in the training set, as shown in Fig. 3(b). The validation accuracy of the 12-feature model fluctuates with varying amount of ternary training samples and exhibits an overall lower accuracy compared to the 24-feature model. Therefore, the inclusion of additional descriptors and ternary training samples results in better generalization of the current model for complex intermetallics. A comprehesive comparsion of 12-feature models and 24-feature models that are trained with pure binary intermetallics, pure ternary intermetallics, and a combination of both is listed in Table I in the Appendix.

Model No. . | Number of descriptors^{a}
. | Training data . | Number of training data . | Prediction data . | No. of prediction data . | MAE (eV/atom) . |
---|---|---|---|---|---|---|

1 | 12 | Binary (80%) | 1230 | Binary (20%) | 424 | 0.044 |

Ternary | 2118 | 0.087 | ||||

2 | 12 | Ternary (80%) | 1694 | Ternary (20%) | 424 | 0.073 |

3 | 12 | [Binary + ternary] (80%) | 2925 | [Binary + ternary] (20%) | 731 | 0.060 |

4 | 24 | Binary (80%) | 1230 | Binary (20%) | 424 | 0.041 |

Ternary | 2118 | 0.057 | ||||

5 | 24 | Ternary (80%) | 1694 | Ternary (20%) | 424 | 0.052 |

6 | 24 | [Binary + ternary] (80%) | 2925 | [Binary + Ternary] (20%) | 731 | 0.046 |

Model No. . | Number of descriptors^{a}
. | Training data . | Number of training data . | Prediction data . | No. of prediction data . | MAE (eV/atom) . |
---|---|---|---|---|---|---|

1 | 12 | Binary (80%) | 1230 | Binary (20%) | 424 | 0.044 |

Ternary | 2118 | 0.087 | ||||

2 | 12 | Ternary (80%) | 1694 | Ternary (20%) | 424 | 0.073 |

3 | 12 | [Binary + ternary] (80%) | 2925 | [Binary + ternary] (20%) | 731 | 0.060 |

4 | 24 | Binary (80%) | 1230 | Binary (20%) | 424 | 0.041 |

Ternary | 2118 | 0.057 | ||||

5 | 24 | Ternary (80%) | 1694 | Ternary (20%) | 424 | 0.052 |

6 | 24 | [Binary + ternary] (80%) | 2925 | [Binary + Ternary] (20%) | 731 | 0.046 |

^{a}

The descriptors in the 12-feature models are mean, maximum, and minimum attributes of number of valence electrons (VEC), 1st ionization energy (FIE), electronegativity (EN), and cohesive energy (CE). The descriptors in 24-feature models contain, mean, maximum, and minimum attributes of molar volume (MV), melting temperature (MT), bulk modulus (BM) and atomic radius (AR), in addition to those in the 12-feature models.

Moreover, considering the scarcity of complex intermetallics with four or more elements, we implemented an adaptive transfer learning algorithm based on Gaussian processes (AT-GP) and tested with ternary intermetallics.^{37} The AT-GP algorithm aims to transfer the shared knowledge from a previously learned GPR model (source task) to other related tasks (target task) by measuring the similarity between the tasks through a semi-parametric transfer kernel. Transfer learning is especially valuable for overcoming the problem of limited data in materials science. It has been recently used for structure–property predictions and to discover new materials with targeted properties such as solid lithium-ion conductors and new piezoelectrics.^{46–51} We use binary intermetallics as the source task and ternary intermetallics as the target task. By varying the amount of ternary training samples in the target task and making prediction on all 2118 ternary intermetallics, we compare the learning curve of the AT-GP model with a No-transfer model that uses only the target task (ternary intermetallics) as training samples, and a Transfer-all model that uses both the source task and target task as training samples. The comparison between the three models is shown in Fig. 7. As expected, the Transfer-all scheme exhibits better overall accuracy with more target samples. However, when limited ternary data are included in the target task (< ∼ 300 samples or ∼15% of the ternary dataset), the AT-GP model gives better performance, as shown in the shaded area in Fig. 7. It also requires less training time. In future, a similar process can be used for extending our model to predict quaternary and more complex intermetallics.

We combined the dataset of ternary intermetallics with that of the binary intermetallics and trained the 24-feature model with 80% of the entries in the combined dataset, and evaluated with the remaining 20%. From the learning curve, as shown in Fig. 6(b), we obtain a MAE of 0.044 ± 0.002 eV/atom on the testing set. We found that the model-predicted Δ*H _{f}* values agree well with those calculated using DFT for both binary and ternary intermetallics, which are plotted seperately in Fig. 3(c). This observed improvement in accuracy is due to the more than twofold increase in the amount of training data. The histograms of absolute error for binary and ternary intermetallics are shown in Figs. 3(d) and 3(e), respectively. We propose that, by similar conditioning to a limited set of multielement intermetallic systems, the present model could be extended to predict the formability of intermetallics from a compositionally vast space involving multiple elements (4, 5, or even more), whose Δ

*H*are rarely available from first-principles-based databases. The performance of our model in terms of MAE in the predicted Δ

_{f}*H*is comparable to other recently proposed models: ElemNet, whose training set size was 230 960 compounds, has a MAE of 0.055 eV/atom on the testing set;

_{f}^{52}CGCNN, which consists of both compositional and structural descriptors with 28 046 training data, has a MAE of 0.039 eV/atom on the testing set.

^{53}

Having established the capability of the model to accurately capture *ΔH _{f}* of known binary and ternary intermetallics, we used it to explore the possibility of finding new binary intermetallics that have not been reported. We screened all binary combinatorial compositions of the 48 metallic elements, which results in 1128 unique pairs with 603 of them reported in the Materials Project. For each pair of elements, whether stable intermetallic compounds have been reported or not, a list of binary compositions ranging from $A0.1B0.9toA0.9B0.1$ (10% mole concentration intervals) is transferred into numerical attributes. We then use the developed model to predict their

*ΔH*and construct a convex hull. The lowest

_{f}*ΔH*on the convex hull is recorded. For those binary intermetallic systems that have been reported, the predicted convex hull agrees well with DFT results from the Materials Project. As an example, the convex hull of Ti–Al predicted by the model is compared to that obtained from the Materials Project in Fig. 8. The absolute error of the predicted lowest

_{f}*ΔH*with respect to the DFT-calculated value for the reported 603 binary pairs can be visualized in the colormap in Fig. 4(a). 80% of the known binary intermetallics are predicted within an error of 0.025 eV/atom. Furthermore, we predict 112 new metallic pairs to form stable intermetallics with a high probability, as shown in Fig. 4(b). By a high probability, we mean those intermetallics whose bounds for the predicted distribution of

_{f}*ΔH*are negative (since GPR can make probabilistic prediction with meaningful uncertainty). The colormap in Fig. 4(b) indicates the mean prediction value of the lowest

_{f}*ΔH*for each pair of elements. This makes up 21% of pair grids that were blank in the Materials Project, which indicates that there may be more stable binary intermetallic compounds waiting to be explored. We have included a more exhaustive map of ML-predicted combinatorial screening formability in Fig. 9. As suggested by Troparevsky

_{f}*et al.*, the lowest

*ΔH*for each pair of elements can be used as an important tool for assessing the formation of single-phase HEAs.

_{f}^{15}

One of the binary combinations that does not have any stable intermetallics reported in the Materials Project is Nb-V, which is shown as a horizontal line (blue) in the phase diagram in Fig. 5(a). The combination is labeled in Fig. 4(b) with a green star. Nb and V alloys have an miscibility gap below 500 K.^{54} However, our model predicts a stable intermetallic for Nb_{0.3}V_{0.7} with a *ΔH _{f}* of −0.097 eV/atom and a credible interval of 0.060 eV/atom, which suggests with a 95% probability that its

*ΔH*should range between −0.157 and −0.037 eV/atom. To confirm this prediction, we performed DFT calculations. As our model does not capture crystal structures, we screened structure prototypes that are formed by refractory metals (Nb, Zr, Ti, and V) near this composition on the Inorganic Crystal Structure Database (ICSD) and the Materials Project.

_{f}^{55}The structures and compositions we found in ICSD are listed in Table II in the Appendix. We then calculated their

*ΔH*using DFT by substituting Nb and V into these structures; some of these are shown in Fig. 5(b). We found two Laves structures, C15 and C14 to be thermodynamically stable, with

_{f}*ΔH*values of −0.045 eV/atom and −0.059 eV/atom, respectively, calculated with respect to bcc V and hcp Nb. In both of the structures, V atoms form tetrahedra around Nb with Nb atoms ordered either in a diamond cubic structure (C15-Laves) or in a hexagonal structure (C14-Laves). To further confirm the dynamical stability of these Laves phases, we performed phonon calculations. We did not observe any soft modes in their phonon band structure plots shown in Figs. 5(c) and 5(d).

_{f}Composition . | ICSD-ID . | MP-ID . | Structure type . | Space group . |
---|---|---|---|---|

(Nb, Ti) | 105248 | NaN | bcc-W | Im-3 m (229) |

Ti_{3}Nb | 671498 | mp-980945 | Auricupride-Cu_{3}Au | Pm3 m (221) |

Ti_{3}Nb | NaN | mp-1187514 | Hexagonal | P6_{3}/mmc (194) |

TiAl_{3} | 58189 | NaN | TiAl_{3} | I4/mmm (139) |

V_{2}Zr | 106214 | mp-258 | Laves(cub)-MgCu_{2} | Fd-3 m (227) |

V_{2}Zr | 653414 | NaN | Laves(2H)-MgZn_{2} | P6_{3}/mmc (194) |

ZrTi_{2} | 247962 | mp-1008568 | CaHg_{2} | P6/mmm (191) |

Composition . | ICSD-ID . | MP-ID . | Structure type . | Space group . |
---|---|---|---|---|

(Nb, Ti) | 105248 | NaN | bcc-W | Im-3 m (229) |

Ti_{3}Nb | 671498 | mp-980945 | Auricupride-Cu_{3}Au | Pm3 m (221) |

Ti_{3}Nb | NaN | mp-1187514 | Hexagonal | P6_{3}/mmc (194) |

TiAl_{3} | 58189 | NaN | TiAl_{3} | I4/mmm (139) |

V_{2}Zr | 106214 | mp-258 | Laves(cub)-MgCu_{2} | Fd-3 m (227) |

V_{2}Zr | 653414 | NaN | Laves(2H)-MgZn_{2} | P6_{3}/mmc (194) |

ZrTi_{2} | 247962 | mp-1008568 | CaHg_{2} | P6/mmm (191) |

The combination of our ML model with DFT calculations can accelerate the discovery of new, compositionally complex intermetallics. As a general strategy, we propose that, for a given combination of elements, our model can be used to rapidly identify compositions that are predicted to form stable intermetallics with a high probability. One can then substitute the elements in the identified intermetallic compositions with chemically similar elements, for instance, by using the probabilistic model developed by Hautier *et al*.,^{56,57} to obtain a set of similar intermetallics, with the expectation that some of them may have been reported previously. This should be followed by a search in the existing databases to screen the various crystal structures adopted by the identified intermetallics. Subsequently, DFT calculations should be used to optimize the targeted composition with the screened structures and determine their thermodynamic stability.

## CONCLUSIONS

We have developed a fast and accurate ML model to predict the formability of binary/ternary intermetallics given any metallic pair/triplet combination. The model achieves strong performance within the range of metallic alloys. Our 12-feature model for binary intermetallics predicts the formation enthalpy of the testing set with a MAE of 0.044 eV/atom.

Our model enables the screening of millions of compositions within seconds, which is ideal for exploring the vast combinatorial space for compositionally complex intermetallics. Meanwhile, we can use the current model developed for binary and ternary intermetallics to predict the stability of intermetallic phases in MPEAs. A statistical study on 142 intermetallic-containing MPEAs has shown that all of the intermetallic phases contained in the MPEAs are existing structures in the binary/ternary subsystems of their respective alloys.^{58} Therefore, by predicting *ΔH _{f}* for a list of binary/ternary compositions formed by each pair/triplet of the constituent elements with our model, and quantitatively comparing with the configurational entropy of the multi-element system at different temperatures, we can predict whether the stable phase of MPEAs would be an ordered structure, disordered solid solution, or a combination of both. Instead of directly searching the combinatorial space of five or more elements, our model provides another prospective method to predict the intermetallic phases likely to form in MPEAs.

To facilitate the discovery of new intermetallics, we are making our model available at https://github.com/M-cube-wustl/ML_intermetallics.

## ACKNOWLEDGMENTS

This work was supported by the National Science Foundation (NSF) through Grant No. DMR-1809571. This work used the computational resources of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF ACI-1548562. The authors are grateful to Professor Rampi Ramprasad and Dr. Logan Ward for fruitful discussions.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

### APPENDIX

Learning curves for the 12-feature model and 24-feature model are shown in Fig. 6. Comparison of the adaptive transfer model (AT-GP) with a model having a No-transfer scheme and one with a Transfer-all scheme is shown in Fig. 7. Model-predicted convex-hull diagram for Ti–Al is shown in Fig. 8. Model-predicted lowest formation enthalpy for each pair of elements is shown in Fig. 9. A summary of the performance of models with different amount of training data and descriptors is shown in Table I. Structures used for searching the crystal structure of predicted Nb-V intermetallic is shown in Table II.

## REFERENCES

*Adaptive Transfer Learning*(AAAI Publications, 2010).

**18**(40),