Machine learning surrogate models for strain-dependent vibrational properties and migration rates of point defects

Machine learning surrogate models employing atomic environment descriptors have found wide applicability in materials science. In our previous work, this approach yielded accurate and transferable predictions of the vibrational formation entropy of point defects for $\mathcal{O}\left(N\right)$ computational cost. The present study investigates the limits of data driven surrogate models in accuracy and applicability for vibrational properties. We propose an improvement of the accuracy by extending the fitting capacity of the model by increasing the dimension of the descriptor space. This is achieved by using a nonlinear relation between descriptors—target observables and when it is possible by including physical relevant information of the underlying energy landscape. The nonlinear extension is used to learn the formation entropy of defects with or without applied strain while including physical information, such as the minimum-saddle point sequences employed for the migration of point defects, is a key ingredient of transition state theory rate approximations. We find excellent predictive power after augmenting the dimensionality of the descriptor space, as demonstrated on large defect databases in $\alpha$-iron and amorphous silicon based on semiempirical force fields. The current linear surrogate models are used to investigate the correlation between migration entropy and energy. Our approaches reproduce the Meyer-Neldel compensation law observed from direct calculations in amorphous Si systems. Moreover, the same abstract descriptor space representation for entropy and energy is then used for the statistical correlation analysis. For linear surrogate models, we show that the energy-entropy statistical correlations can be reinterpreted in descriptor space. This provides a simple statistical criterion for the marginal interpretation of the compensation law. More generally, the present work shows how linear surrogate models can accelerate high-throughput workflows, aid the construction of mesoscale material models, and provide new avenues for correlation analysis.

Figure 1

Graphical abstract of the vibrational entropy calculations using the traditional (a) and the present workflows of vibrational harmonic entropies. Atomic configurations are generated and optimized using the artn method and semiempirical potentials for $\alpha$-Fe and amorphous Si. Then, in the traditional approach (a), the system's Hessian matrix is computed and diagonalized in order to obtain vibrational harmonic entropy $S^{\text{vib}}$. In the present machine learning workflow (b) to estimate $S^{\text{vib}}$, we calculate the atomic descriptors of the same atomic configurations. Then the surrogate regression model is trained using the corresponding database of minimum and saddle point configurations.

Figure 2

Illustration of the performance of linear and EQML surrogate models using deformed supercells of $I_{2-4}/V_4$ clusters and by using $b\mathrm{SO}{\left(4\right)}_4$ descriptor. The initial configurations have a ${\left(8a_0\right)}^3$ volume and have been deformed by applying an homogeneous and isotropic dilatation of the supercell. The deformation rates range from $-1\%$ to $3\%$. Figures illustrate the performances of linear (a) and EQML (b) of surrogate models.

Figure 3

Illustration of the linear model to adjust the (ln) attack frequencies of the amorphous Si database, where ${\nu }_0=1$ THz. The color gradient represents the data distribution, with yellow corresponding to dense data zones. (a) shows the regression of the logarithm of attack frequencies; the results of the train/test validation procedure are presented in the inset. The values of statistical indicators remain stable, even for a large proportion of validation sets. (b) emphasizes the results of attack frequencies. The logarithm of noise $\text{ln}(\nu /\widehat{\nu })$ ($\widehat{\nu }$ is the frequency estimation using the surrogate model of the direct computed frequency $\nu$ from ML model) is presented as inset of (b) and follows a normal distribution with mean $\mu =0$ and standard deviation $\sigma =0.653$.

Figure 4

The correlation between the predicted and computed energy barrier $\mathrm{\Delta }E$ using the linear regression between descriptors and energy barriers on the amorphous Si database. The relative accuracy of the linear regression model is better for barriers than for the log of prefactors (a RMSE of 0.09 eV over a range of 6 eV).

Figure 5

Drawing the enthalpy-entropy compensation relation for the Si amorphous database. (a) shows values of $\mathrm{\Delta }E$ and $\text{ln}(\nu /{\nu }_0)$ computed with artn method. (b) shows predicted values of $\mathrm{\Delta }E$ and $\text{ln}(\nu /{\nu }_0)$ with linear model. The color gradient represents data distribution; yellow corresponds to dense data zones for both types of points. Adjusted EEC relations, following Eq. (29), for both direct and surrogate models are emphasized in (a) and (b) by white points with black contour. Both models, the direct and the surrogate, have distributions with very close correlation indicators. Marginal variance distribution for both dataset is presented in (c) and (d) for direct and surrogate data respectively. The marginal variance is quantitatively almost the same for both dataset.