Standard Deviation Effect of Average Structure Descriptor on Grain Boundary Energy Prediction

The structural complexities of grain boundaries (GBs) result in their complicated property contributions to polycrystalline metals and alloys. In this study, we propose a GB structure descriptor by linearly combining the average two-point correlation function (PCF) and standard deviation of PCF via a weight parameter, to reveal the standard deviation effect of PCF on energy predictions of Cu, Al and Ni asymmetric tilt GBs (i.e., Σ3, Σ5, Σ9, Σ11, Σ13 and Σ17), using two machine learning (ML) methods; i.e., principal component analysis (PCA)-based linear regression and recurrent neural networks (RNN). It is found that the proposed structure descriptor is capable of improving GB energy prediction for both ML methods. This suggests the discriminatory power of average PCF for different GBs is lifted since the proposed descriptor contains the data dispersion information. Meanwhile, we also show that GB atom selection methods by which PCF is evaluated also affect predictions.


Introduction
Grain boundaries (GBs) are one of the most commonly seen planar defects in polycrystalline metals and alloys. Due to local atomic distortions and inconsistent atomic arrangement, GBs play important roles in determining the mechanical, thermal and electric, etc., properties of materials [1,2]. For example, GBs may act as the dislocation and point defect sources or sinkers, and they may block the dislocation motion and absorb them; thus the strength and ductility of materials can be greatly changed [3]. For an idealized GB, from the geometrical point of view, it can be completely governed by five parameters, usually represented by misorientation and a normal GB plane [4]. Unfortunately, the structures, as well as the properties of the GB, are hard to completely determine. This is simply because a GB may have numerous states due to the point defect absorptions and emissions. It means the structures of a given GB may no longer be unique for a given energy [5][6][7][8][9][10][11]. Thus, the connection between structure and property of a GB, such as energy, volume and mechanical behavior, etc., are usually built via atomistic simulations by using molecular dynamics (MD) and density functional theory (DFT) methods [12,13], which is also of great significance for the macroscopic modeling of material behavior [14,15]. Technically speaking, it is possible to do so using MD and DFT, but also needs a heavy workload if such connections for a large number of GBs are expected.
The ML method has been applied in many research fields [16][17][18][19][20], and provides an efficient technique by which to link the structure-property of a GB, particularly to extract correlations from high-dimensional datasets [21][22][23][24][25], and has been successfully applied in predicting GB energies [21,[23][24][25][26][27], point defect segregation energies [28,29], GB structures [30] and damages and deformations in GB [31,32]. Usually, an appropriate ML method is employed according to the datasets and the expected correlations. Regardless of these, a problem is how to mathematically describe the GB structure, which should contain regression of energies of Cu, Al and Ni GBs according to the tilt axis of GB considering full data and partition data is discussed. Thirdly, the effect of standard deviation of PCF on the RNN-based prediction of GB energies is discussed. Finally, the prediction comparisons between the two ML methods and conclusions of this study are made.

Methodology and GB Structure Descriptor
We consider a total number of 464 asymmetric tilt GBs (ATGBs) with misorietations Σ3, Σ5, Σ9, Σ11, Σ13 and Σ17 as the dataset for the subsequent GB energy prediction study. Each GB model is a bi-crystal composed of two grains with specified orientations. To construct it with periodic boundary conditions (PBCs) applicable, crystalline orientations of two grains are needed. Usually, a given GB misorientation Σ can be defined by the overlapped lattices of two crystals with one of them rotated around a specified axis ρ with a certain angle θ. Namely, Σ is equivalent to (ρ, θ), which can also be represented by a rotation matrix R Σ . Take Σ = 3 as an example, (ρ, θ) related to Σ3 equals ([110], 70.53 • ) [33], and the corresponding unit cell of Σ3 coincidence site lattice (CSL) is spanned by [ 10], the crystal orientations of two grains of all Σ3 ATGBs can be defined. By following this approach, asymmetric GBs of all other misorientations can be readily created.
For Σ3, Σ5, Σ9, Σ11, Σ13 and Σ17, lattice symmetry requires the inclination angle φ varying from 0 • to 90 • for Σ3, Σ9 and Σ11, with φ varying from 0 • to 45 • for Σ5, Σ13 and Σ17, respectively. PBCs are imposed within the GB plane for all bicrystal models. Two grains (i.e., Grains A and B) terminate with free surfaces in the direction perpendicular to the GB plane by setting two 10Å thick vacuum spaces on the top and bottom ends of the bicrystal model, as schematically shown in Figure S1 in Supplementary Materials. This allows us to release the stress possibly produced in the z direction during the GB structure optimization. Embedded atom method (EAM) potentials [45,46] are used to model the atomic interactions in Al [47], Cu [48] and Ni [47]. We relax all 464 ATGBs via the conjugate gradient (CG) method using LAMMPS [49] and compute the average energies of all GBs. Tables S1 and S2 in Supplementary Materials list the variations in atom numbers and energies for Σ3, Σ5, Σ9, Σ11, Σ13 and Σ17 GB models of each metal. Atomic structures are visualized using Ovito [50].
With all GBs relaxed, we are in a position to introduce the descriptor by which the GB structure and structure differences between GBs can be described and distinguished. Herein, we employ the pair correlation function (PCF) method proposed Gomberg et al. [21] as a GB structure descriptor. In doing so, a primary concern is how many atoms in GB should be considered when evaluating the average PCF (PCF mean (r)). In other words, an appropriate method for selecting GB atoms should include a certain number of atoms in the vicinity of the GB carrying structure information, but exclude other atoms. Herein, we consider common neighbor analysis (CNA) [43] and centro-symmetric parameter (CSP) methods [44]. For the CSP method, two CSP critical values are considered (i.e., CSP > 0.1 and 0.5). As exemplified in Figure S2 in Supplementary Material, the three methods identify a different number of GB atoms. Such effects on GB energy predictions will be discussed in the following section.
According to the approach of Gomberg et al. [21], the PCF of a given GB is computed by averaging the radial distribution function g a (r) of all N GB GB atoms selected out of the GB using CNA, CSP 0.1 and CSP 0.5 , which can be expressed as PCF mean (r) = 1 N GB ∑ α∈GB g a (r) (1) where the radial distribution function g a (r) of a GB atom can be calculated by considering all of its N in neighboring atoms within a specified cut-off radius for three metals.
where a kernel function K e with bandwidth h e is used to smoothen the radial distribution function. n 0 is the atom density in the FCC lattice and R k α is the distance between atom a and its kth neighboring atom. The parameters needed in Equation (2) are listed in Table S3 in Supplementary Material. Figure 1a compares PCF mean of the Al single crystal with results taken from [21]. Good agreement validates our algorithm for computing PCF mean . In fact, PCF mean is an averaged radial distribution function (RDF) curve of each GB atom, by which the PCF data fluctuations of different GB atoms cannot be well considered, as evidenced by the variation in standard deviation of PCF (PCF std (r)) for three GBs in Cu in Figure 1b. In order to incorporate the data fluctuation into the averaged PCF, we further propose a PCF comb by combining PCF mean (r) and PCF std (r) as where parameter ζ is introduced to weigh the portions of PCF mean (r) and PCF std (r) in PCF comb . PCF comb is reduced to PCF by letting ζ = 0. As an example, Figure 1c,d shows the PCF comb (r) of ∑5(310) and ∑9(114) Cu GBs for three values of ζ. Clearly, the variation trends of PCF comb (r) changes as ζ varies. In the following, how the variation of ζ influences the prediction will be discussed. The PCF comb curve of each GB is further represented as 512 discrete points, serving as the input data for the ML methods.
In the following, we adopt two ML methods to predict GB energies, i.e., principal component analysis (PCA)-based linear regression [40] and recurrent neural networks (RNN) [41,42]. PCA is usually implemented in two steps, a dimensionality-reduction of data and regression based on the principle component, which are essentially the eigenvalues of the covariance matrix of raw data. Therefore, the regression of PCA is achieved only using a few principle components. The principle components for regression are selected by considering the explained variance percentage of each principle component. However, RNNs do not require dimensionality reduction. The training and prediction are performed by using raw data. To quantitatively compare the predictions, mean absolute error (MAE) and mean relative error (MRE) are assessed via where γ Pred where parameter is introduced to weigh the portions of PCFmean(r) and PCFstd(r) in PCFcomb. PCFcomb is reduced to PCF by letting = 0. As an example, Figure 1c,d shows the PCFcomb(r) of 5(310) and 9(114) Cu GBs for three values of . Clearly, the variation trends of PCFcomb(r) changes as varies. In the following, how the variation of influences the prediction will be discussed. The PCFcomb curve of each GB is further represented as 512 discrete points, serving as the input data for the ML methods. In the following, we adopt two ML methods to predict GB energies, i.e., principal component analysis (PCA)-based linear regression [40] and recurrent neural networks (RNN) [41,42]. PCA is usually implemented in two steps, a dimensionality-reduction of data and regression based on the principle component, which are essentially the eigenvalues of the covariance matrix of raw data. Therefore, the regression of PCA is achieved only using a few principle components. The principle components for regression are selected by considering the explained variance percentage of each principle component. However, RNNs do not require dimensionality reduction. The training and prediction are performed by using raw data. To quantitatively compare the predictions, mean absolute error (MAE) and mean relative error (MRE) are assessed via where and are GB energies predicted by ML methods and computed via MD.

PCA-Based Prediction
To implement PCA, we need to determine which principle components will be used in the regression. To do so, we analyze the explained variance percentage of the first ten principle components for Cu, as shown in Figure S4 in Supplementary Materials. It turns out that the explained variation of the first PC is up to 93%, while those of the other nine PCs are lower than 3%. This suggests that only the first few PCs accounting for higher explained variance percentages retain most of the original data, while the rest only keep a small amount of the data. It is therefore unnecessary to consider many PCs in the sub-

PCA-Based Prediction
To implement PCA, we need to determine which principle components will be used in the regression. To do so, we analyze the explained variance percentage of the first ten principle components for Cu, as shown in Figure S4 in Supplementary Materials. It turns out that the explained variation of the first PC is up to 93%, while those of the other nine PCs are lower than 3%. This suggests that only the first few PCs accounting for higher explained variance percentages retain most of the original data, while the rest only keep a small amount of the data. It is therefore unnecessary to consider many PCs in the subsequent GB energy regression. Because of this, only the first three PCs, e.g., PC 1 , PC 2 and PC 3 , are used in the regression. With the multiple linear regression method, GB energy can be written as where a, b, c and d are fitting parameters. PC 1 , PC 2 and PC 3 are obtained from data training, which is dependent on the dataset. The dataset in this study consists of GBs with <100> and <110> tilts axes. Thus, there are two possible ways to obtain PCs by reducing the dimensionality of data when considering all data together and two data subsets corresponding to <100> and <110> tilt axes, denoted as full data and partitioned data methods, respectively. Thus, two sorts of PCs PC i through training different datasets can be obtained. The PCs obtained from data training are actually a representation of data in a lower dimensional space. Although these PCs retain most of the original data, it is still challenging to impart each PC with possible physical interpretability. By following the approach by Gomberg et al. [14], it is possible to correlate each PC to a geometrical parameter of GB by interpolating each PC as a function of the geometrical parameter. Herein, such a geometrical parameter is considered to be inclination angle φ related to asymmetric GBs for a specified Σ. In this study, PC 1 , PC 2 and PC 3 are assumed to be cubic polynomial interpolation functions of φ.
where A i , B i , C i and D i are fitting parameters. Such cubic polynomial interpolation can well characterize the variation of calculated PC i vs φ for most GBs and PCs. From the above analysis, there are four ways of predicting GB energies in terms of different approaches of obtaining PCs, denoted as full data, full data-fitting, partitioned data and partitioned data-fitting methods, respectively. For the partitioned data, Equation (4)  From further inspection of the variation of MAEs due to the data partition, MAEs for PCi-and PCi(φ)-based linear fittings are reduced by~30% and~23%, respectively, but, for Σ3 GBs, they are up to~83% and~64%. This suggests that a better prediction can be achieved by separately considering <110> and <100> GB datasets, which is particularly more prominent for <100> GBs. From the MAEs results of PCi and PCi(φ) linear fittings for Σ3 and Σ5 GBs, PCi(φ) linear fittings indeed lead to a larger MAE than PCi linear fittings, which is understandable since PCi(φ) is approximately obtained from cubic interpolation. Nevertheless, these results show that PCi is capable of being correlated with inclination angle φ.
As previously mentioned, γ Pred i should be dependent on ζ, therefore, both MAE and MRE are functions of ζ. As an example, Table 1 shows MAEs and MREs of all Σs for ζ = 0.5. Comparing the MAE and MRE predictions for <100> and <110> GBs, both MAE and MRE are lower for <100> GBs, which further demonstrates that better predictions can be obtained for <100> GBs. In fact, this can be explained by considering the structure differences of <110> and <100> GBs. It is known that SUs for <100> GBs are composed of some [100] dislocations [2,51,52]. This brings simpler and mutually similar structures to <100> GBs. However, <110> GBs are composed of SUs much more complicated than those of <100> GBs [2,33,[53][54][55]. Therefore, the structures of two <100> GBs may be quite different from each other. Thus, predictions for <100> GBs are better than those for their <110> counterparts, as also evidenced by the results of Al and Ni (see Figures S11, S12, S16 and S17 in Supplementary Material).  As previously mentioned, should be dependent on , therefore, both MAE and MRE are functions of . As an example, Table 1 shows MAEs and MREs of all Σs for = 0.5. Comparing the MAE and MRE predictions for <100> and <110> GBs, both MAE and MRE are lower for <100> GBs, which further demonstrates that better predictions can be obtained for <100> GBs. In fact, this can be explained by considering the structure differences of <110> and <100> GBs. It is known that SUs for <100> GBs are composed of some [100] dislocations [2,51,52]. This brings simpler and mutually similar structures to <100> GBs. However, <110> GBs are composed of SUs much more complicated than those of <100> GBs [2,33,[53][54][55]. Therefore, the structures of two <100> GBs may be quite different from each other. Thus, predictions for <100> GBs are better than those for their <110> counterparts, as also evidenced by the results of Al and Ni (see Figures S11, S12, S16 and S17 in Supplementary Material).    (Figures (a,c)) and partitioned data (Figures (b,d)). Note that this figure exemplifies the predictions using PCF comb computed for CNA-based GB atom selection and ζ = 0.5. Results of Σ9, Σ11, Σ13 and Σ17 GBs based on full data and partitioned data are shown in Figures S5-S8 in Supplementary Material.
In order to compare the effects of the GB atom selection method (i.e., CNA, CSP = 0.1 and CSP = 0.5) on the prediction, Figure 3a,b exemplify the MRE of <110> and <100> Cu GBs vs. ζ. From Figure 3, with increasing ζ, the MRE of <110> GBs keep increasing; however, that of <110> GBs keep decreasing. Finally, MREs of both <110> and <100> GBs reach plateaus. Further inspection of Figure 3 reveals that the minimum values of MRE for <110> and <100> GBs for CAN and CSP0.1 methods correspond to ζ = 0.0 and 1.0, respectively. For the CSP0.5 method, the minimum values of MRE for <110> and <100> GBs are ζ = 0.2 and 0.1. Moreover, considering CAN, CSP0.1 and CSP0.5 alone, MREs also differ at ζ = 0.0, but their general variation trends are similar. Therefore, it can be seen that a better prediction not only requires an appropriate GB atom selection method, but an appropriate value of ζ. In fact, the MREs of Al and Ni are also dependent on ζ, as seen from Figures S15 and S20 in Supplementary Material. for <110> and <100> GBs for CAN and CSP0.1 methods correspond to = 0.0 and 1.0, re-spectively. For the CSP0.5 method, the minimum values of MRE for <110> and <100> GBs are = 0.2 and 0.1. Moreover, considering CAN, CSP0.1 and CSP0.5 alone, MREs also differ at = 0.0, but their general variation trends are similar. Therefore, it can be seen that a better prediction not only requires an appropriate GB atom selection method, but an appropriate value of . In fact, the MREs of Al and Ni are also dependent on , as seen from Figures S15 and S20 in Supplementary Material.

RNN-Based Predictions
In this section, we discuss the predictions using the RNN method. Due to the dimension reduction in the PCA method, some data is lost. Moreover, a better prediction can be achieved provided that GB types are distinguished based on the tilt axis; i.e., <100> and <110>. In comparison to PCA, on the other hand, the RNN method is highly nonlinear. Considering all of these factors, we do not distinguish GB types for each metal; i.e., the prediction is performed using the full data for each metal. For each metal, 10-fold cross validation is performed, with each fit being performed on a training set consisting of 70% of the total training set selected at random, with the remaining 30% used as a holdout set for testing. This yields good convergence of MAE, as shown in Figure S21 in Supplementary Materials. Figure 4 shows the prediction results of the RNN method considering three GB atom selection methods. A preliminary comparison between PCA and RNN, as shown in Figures 2 and 4, reveals that the RNN method gives a better prediction. This is not surprising due to the higher nonlinearity of the RNN method.

RNN-Based Predictions
In this section, we discuss the predictions using the RNN method. Due to the dimension reduction in the PCA method, some data is lost. Moreover, a better prediction can be achieved provided that GB types are distinguished based on the tilt axis; i.e., <100> and <110>. In comparison to PCA, on the other hand, the RNN method is highly nonlinear. Considering all of these factors, we do not distinguish GB types for each metal; i.e., the prediction is performed using the full data for each metal. For each metal, 10-fold cross validation is performed, with each fit being performed on a training set consisting of 70% of the total training set selected at random, with the remaining 30% used as a holdout set for testing. This yields good convergence of MAE, as shown in Figure S21 in Supplementary Materials. Figure 4 shows the prediction results of the RNN method considering three GB atom selection methods. A preliminary comparison between PCA and RNN, as shown in Figures 2 and 4, reveals that the RNN method gives a better prediction. This is not surprising due to the higher nonlinearity of the RNN method.
We further evaluated the MAE of RNN predictions for three GB atom selection methods and three metals, as shown in Figure 5. Clearly, with increasing ζ, there is a sudden drop in the MAE. Meanwhile, such a drop in CNA, CSP0.1 and CSP0.5 for the same metal almost occurs at the same value of ζ. Moreover, MAEs for Cu, Al and Ni suddenly drop by~75%,~75% and~70% at ζ crit ≈ 0.3, 0.6 and 0.7. After the sudden drops, all curves approach plateaus with nearly the same MAE, regardless of the three GB atom selection methods. This evidences the significant dependence of ζ in RNN prediction, and also implies that considerable errors will be caused in RNN prediction when letting ζ = 0. To avoid such errors, the standard deviation of PCFs (PCF std ) must be incorporated into the descriptor. Moreover, the ζ crit of the three metals differ, which implies that the prediction accuracy can be enhanced only when more data scatter information of PCF is considered. ζ crit of Cu is the smallest, while that of Ni is the largest. It suggests that data scatter of the average PCF for Cu is the lowest, but that of Ni is the highest. of 70% of the total training set selected at random, with the remaining 30% used as a holdout set for testing. This yields good convergence of MAE, as shown in Figure S21 in Supplementary Materials. Figure 4 shows the prediction results of the RNN method considering three GB atom selection methods. A preliminary comparison between PCA and RNN, as shown in Figures 2 and 4, reveals that the RNN method gives a better prediction. This is not surprising due to the higher nonlinearity of the RNN method.  We further evaluated the MAE of RNN predictions for three GB atom selection methods and three metals, as shown in Figure 5. Clearly, with increasing , there is a sudden drop in the MAE. Meanwhile, such a drop in CNA, CSP0.1 and CSP0.5 for the same metal almost occurs at the same value of . Moreover, MAEs for Cu, Al and Ni suddenly drop by ~75%, ~75% and ~70% at crit≈0.3, 0.6 and 0.7. After the sudden drops all curves approach plateaus with nearly the same MAE, regardless of the three GB atom selection methods. This evidences the significant dependence of in RNN prediction and also implies that considerable errors will be caused in RNN prediction when letting =0. To avoid such errors, the standard deviation of PCFs (PCFstd) must be incorporated into the descriptor. Moreover, the crit of the three metals differ, which implies that the prediction accuracy can be enhanced only when more data scatter information of PCF is considered. crit of Cu is the smallest, while that of Ni is the largest. It suggests that data scatter of the average PCF for Cu is the lowest, but that of Ni is the highest.  We further evaluated the MAE of RNN predictions for three GB atom selection methods and three metals, as shown in Figure 5. Clearly, with increasing , there is a sudden drop in the MAE. Meanwhile, such a drop in CNA, CSP0.1 and CSP0.5 for the same metal almost occurs at the same value of . Moreover, MAEs for Cu, Al and Ni suddenly drop by ~75%, ~75% and ~70% at crit≈0.3, 0.6 and 0.7. After the sudden drops, all curves approach plateaus with nearly the same MAE, regardless of the three GB atom selection methods. This evidences the significant dependence of in RNN prediction, and also implies that considerable errors will be caused in RNN prediction when letting =0. To avoid such errors, the standard deviation of PCFs (PCFstd) must be incorporated into the descriptor. Moreover, the crit of the three metals differ, which implies that the prediction accuracy can be enhanced only when more data scatter information of PCF is considered. crit of Cu is the smallest, while that of Ni is the largest. It suggests that data scatter of the average PCF for Cu is the lowest, but that of Ni is the highest.

Discussions
In this study, we predicted GB energies by using PCA-based linear regression and RNN. Figure 6 compares the GB energy prediction properties of two ML methods. From Figure 6a, RNN gives a better prediction than the PCA method. For PCA-based linear regression, linear fitting parameters of GB energy are obtained by dividing a full dataset into two separate datasets according to GBs of <100> and <110> tilt axis for three metals. In doing so, we attempted to weaken the effects of the mutual interference due to <100> and <110> GBs when obtaining linear fitting parameters. Indeed, such a way of treating a dataset for PCA-based predictions decreases the prediction errors ( Figure 2). We also tried to further obtain the linear fitting parameters by expressing them as cubic polynomial interpolation functions of an inclination angle of GB ϕ, instead of obtaining them from data training. The purpose of doing so was to obtain an empirical GB energy prediction function and intensify the interpretability of ML prediction, which may be impossible for the RNN method. Such a method may work for PCA-based predictions, as evidenced in Figure 2. An appropriate GB structure descriptor is vital in ML-based GB energy prediction, as whether the descriptor contains the essential information of GB structures or not determines the prediction accuracy. In fact, PCF as a GB structure descriptor [16], compared

Discussions
In this study, we predicted GB energies by using PCA-based linear regression and RNN. Figure 6 compares the GB energy prediction properties of two ML methods. From Figure 6a, RNN gives a better prediction than the PCA method. For PCA-based linear regression, linear fitting parameters of GB energy are obtained by dividing a full dataset into two separate datasets according to GBs of <100> and <110> tilt axis for three metals. In doing so, we attempted to weaken the effects of the mutual interference due to <100> and <110> GBs when obtaining linear fitting parameters. Indeed, such a way of treating a dataset for PCA-based predictions decreases the prediction errors ( Figure 2). We also tried to further obtain the linear fitting parameters by expressing them as cubic polynomial interpolation functions of an inclination angle of GB φ, instead of obtaining them from data training. The purpose of doing so was to obtain an empirical GB energy prediction function and intensify the interpretability of ML prediction, which may be impossible for the RNN method. Such a method may work for PCA-based predictions, as evidenced in Figure 2.

Discussions
In this study, we predicted GB energies by using PCA-based linear regression and RNN. Figure 6 compares the GB energy prediction properties of two ML methods. From Figure 6a, RNN gives a better prediction than the PCA method. For PCA-based linear regression, linear fitting parameters of GB energy are obtained by dividing a full dataset into two separate datasets according to GBs of <100> and <110> tilt axis for three metals. In doing so, we attempted to weaken the effects of the mutual interference due to <100> and <110> GBs when obtaining linear fitting parameters. Indeed, such a way of treating a dataset for PCA-based predictions decreases the prediction errors ( Figure 2). We also tried to further obtain the linear fitting parameters by expressing them as cubic polynomial interpolation functions of an inclination angle of GB ϕ, instead of obtaining them from data training. The purpose of doing so was to obtain an empirical GB energy prediction function and intensify the interpretability of ML prediction, which may be impossible for the RNN method. Such a method may work for PCA-based predictions, as evidenced in Figure 2. An appropriate GB structure descriptor is vital in ML-based GB energy prediction, as whether the descriptor contains the essential information of GB structures or not determines the prediction accuracy. In fact, PCF as a GB structure descriptor [16], compared An appropriate GB structure descriptor is vital in ML-based GB energy prediction, as whether the descriptor contains the essential information of GB structures or not determines the prediction accuracy. In fact, PCF as a GB structure descriptor [16], compared with those of polyhedral units [34,35], is much easier use. However, the definition of PCF shows that the GB structure is described by an average function. It is believed that the GB structure may not be well described without considering the higher order moment of RDF data from the statistical point of view. Motivated by this, we further incorporated the standard deviation of PCF into the PCF function to further extend the descriptor of the GB structure (Equation (2)). Figure 6b shows that predictions using PCA and RNN are significantly dependent on the parameter and GB atom selection methods for three metals, also suggesting the necessity of considering PCF std in GB structure descriptors.

Conclusions
In this paper, we studied the GB energies prediction properties of Cu, Al and Ni using two ML methods; i.e., PCA-based linear regression and RNN. We considered the asymmetric GBs Σ3, Σ5, Σ9, Σ11, Σ13 and Σ17 of <110> and <100> types. Atomistic models were constructed and relaxed using the MD method. By extending the PCF-based GB structure descriptor and using three methods of selecting GB atoms, we compared the prediction of two ML methods. The main conclusions of this study were drawn as: For the three metals, the lowest MAE can be obtained when ζ is greater than 0.8 for RNN, while that should be smaller than 0.3 for PCA-based linear regression. This indicates the dependence of GB descriptors on the ML method. Meanwhile, PCF as an average function and GB structure descriptor needs to consider the PCF std , by which ML prediction accuracy can be improved. The GB structure descriptor in the form of average structure representation (ASR) may need to further take into account the standard deviation of ASR.
In comparison to RNN, it is indeed possible to intensify or realize the interpretability of ML prediction by using the PCF-based linear regression method, though how to generalize the fitting method of the linear regression function when considering a dataset of different GB misorientations still needs to be addressed.
For a specific ML method, the MAE of the prediction is determined by multiple factors, such as the GB atom selection method and a portion of PCF standard deviation. A better quantitative descriptor of GB structure is a trade-off between computation cost and complexity. It is expected that prediction accuracy can be enhanced by combining those comprehensive descriptors together. Moreover, it will be interesting to examine the performance of GB structure descriptors if we consider GBs of mixed types.

Supplementary Materials:
The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/ma16031197/s1, Figure S1. Schematic of the 3D bicrystal GB model. Figure S2. Three methods of selecting GB at-oms in a Σ3 Cu GB. Red atoms in the center of model are those selected out of whole model based on different methods. Figure S3. PCF(r) and PCFcomb(r) of GBs ∑5(310), ∑9(114) and ∑3(111) in Al and Ni. Figure S4. the explained variance percentage for first-ten principle components by considering full data set of Cu. Figure S5. PCA based prediction results of Cu <110> GBs using full data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S6. PCA based prediction results of Cu <100> GBs using full data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S7. PCA based prediction results of Cu <110> GBs using partitioned <110> data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S8. PCA based prediction results of Cu <100> GBs using portioned <100> data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S9. Comparison of Computed PCi with cubic interpolated PCi as a function of inclination angle for <110> type Cu GBs. Figure S10. Comparison of Computed PCi with cubic interpolated PCi as a function of inclination angle for <100> type Cu GBs. Figure S11. PCA based prediction results of Al <110> GBs using partitioned <110> data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S12. PCA based prediction results of Al <100> GBs using portioned <100> data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S13. Comparison of Computed PCi with cubic interpolated PCi as a function of inclination angle for <110> type Al GBs. Figure S14. Comparison of Computed PCi with cubic interpolated PCi as a function of inclination angle for <100> type Al GBs. Figure S15. MRE of Al GBs for three methods of selecting GB atoms. Figure S16. PCA based prediction results of Ni <110> GBs using partitioned <110> data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S17. PCA based prediction results of Ni <100> GBs using portioned <100> data. Regression is performed using PCi based on dimensionality reduction and further interpolation as a function of φ, respectively. Figure S18. Comparison of Computed PCi with cubic interpolated PCi as a function of inclination angle for <100> type Ni GBs. Figure S19. Comparison of computed PCi with cubic interpolated PCi as a function of inclination angle for <100> type Ni GBs. Figure S20. MRE of Cu GBs for three methods of selecting GB atoms. Figure S21. MAE convergence curve for RNN prediction of Cu GBs. Table S1. Some additional information of all GB models. Table S2. Variation range of GB energies for all GBs of each metal (mJ/m 2 ). Table S3. Some material constants and parameter used for computing PCF. Table S4. Cubic interpolation of PCi as a function of inclination angle for all Cu GBs. Table S5. Cubic interpolation of PCi as a function of inclination angle for all Al GBs. Table S6. Cubic interpolation of PCi as a function of inclination angle for all Ni GBs.  Institutional Review Board Statement: Not applicable.

Data Availability Statement:
The data that supports the findings of this study are available in the supplementary material of this article.

Conflicts of Interest:
The authors declare no conflict of interest.