Accelerating the discovery of low-energy structure configurations: a computational approach that integrates first-principles calculations, Monte Carlo sampling, and Machine Learning

TL;DR

This work has leveraged well-established Cluster Expansion techniques with Local Outlier Factor models to establish strategies that enhance the reliability of the CE method and has demonstrated the capabilities of the proposed approach for the particular case of a tungsten-based quaternary high-entropy alloy.

Abstract

Finding Minimum Energy Configurations (MECs) is essential in fields such as physics, chemistry, and materials science, as they represent the most stable states of the systems. In particular, identifying such MECs in multi-component alloys considered candidate PFMs is key because it determines the most stable arrangement of atoms within the alloy, directly influencing its phase stability, structural integrity, and thermo-mechanical properties. However, since the search space grows exponentially with the number of atoms considered, obtaining such MECs using computationally expensive first-principles DFT calculations often results in a cumbersome task. To escape the above compromise between physical fidelity and computational efficiency, we have developed a novel physics-based data-driven approach that combines Monte Carlo sampling, first-principles DFT calculations, and Machine Learning to accelerate the discovery of MECs in multi-component alloys. More specifically, we have leveraged well-established Cluster Expansion (CE) techniques with Local Outlier Factor models to establish strategies that enhance the reliability of the CE method. In this work, we demonstrated the capabilities of the proposed approach for the particular case of a tungsten-based quaternary high-entropy alloy. However, the method is applicable to other types of alloys and enables a wide range of applications.

Figures (5)

• Figure 1: (a) Detailed description of the MC-DFT algorithm. Highlighted is the step where the accelerated MC-DFT (aMC-DFT) is applied; (b) detailed flowchart of the aMC-DFT for calculating the energy of the swapped configuration, a loop that is repeated every MC step.
• Figure 2: Statistical analysis of the hyperparameter optimization for different training sizes: (a, d, g) Average RMSE, $R^2$, and MAE values, respectively, for all combinations (orange color) with green diamonds representing the values for the best hyperparameter combinations. (b, e, h) RMSE, $R^2$, and MAE values, respectively, for the best hyperparameter combinations alongside computational cost (CPU hours). (c, f, i) Predicted vs. DFT energy values produced by a model trained with 300 pairs of structures with their energies $\left(E_i, S_i \right)$ obtained exclusively via MC-DFT, showing the correlation between predicted and actual energy values using the best hyperparameters based on the lowest RMSE, highest $R^2$, and lowest MAE values, respectively.
• Figure 3: Comparison of minimum energy convergence, computational cost, and acceptance rate among different methods during Monte Carlo (MC) simulations: (a) Energy difference ($\Delta E$) as a function of MC steps for MCDFT and a-MCDFT methods. (b) Computational cost (CPU hours) as a function of MC steps for MCDFT and a-MCDFT methods. (c) Cumulative number of accepted states as a function of MC steps for MCDFT and a-MCDFT methods.
• Figure 4: Scalability of the a-MCDFT method on larger $6 \times 6 \times 6$ supercells, containing 432 atoms. The a-MCDFT simulations on the $6 \times 6 \times 6$ supercells used the same hyperparameters obtained from the grid search on the $4 \times 4 \times 4$ supercells. During the first 300 MC steps (training size is one of the hyperparameters of the grid search), the surrogate model was re-trained to obtain a new value of the regression coefficients and the regularization parameter. (a) Energy evolution with respect to the energy of the initial SQS structure, (b) Computational cost (CPU hours), and (c) Cumulative number of accepted states, all of them as a function of MC steps for both MCDFT and a-MCDFT methods on $6 \times 6 \times 6$ supercells.
• Figure 5: Energy evolution per atom (with respect to the energy of the initial SQS structure) predicted by the a-MCDFT approach on both $4 \times 4 \times 4$ and $6 \times 6 \times 6$ supercells. The simulations on the $6 \times 6 \times 6$ supercells used the same hyperparameters obtained from the grid search on the $4 \times 4 \times 4$ supercells. During the first 300 MC steps, the surrogate models is trained at each supercell size to obtain the most accurate values of the regression coefficients and the regularization parameter.