Machine Learning for Improved Density Functional Theory Thermodynamics

TL;DR

This work tackles the limited energy resolution of density functional theory (DFT) in predicting alloy phase stability by introducing Error Corrected DFT (EC-DFT), a machine-learning framework that learns the correction $H_{ ext{corr}} = H_{ ext{DFT}} - H_{ ext{expt}}$ to formation enthalpies. A structured feature set (concentrations, weighted atomic numbers, and interaction terms) feeds both a linear model and a multi-layer perceptron (MLP) regressor, with the latter delivering substantially lower RMSEs via cross-validation on binary and ternary systems. Applied to the Al-Ni-Pd and Al-Ni-Ti ternaries, the neural network achieves RMSEs as low as ~5.5 meV/atom overall and ~10 meV/atom on unseen compositions, significantly improving DFT-based formation enthalpy predictions. The EC-DFT framework provides a scalable, interpretable route to more reliable phase-diagram predictions, aiding accelerated computational materials design while highlighting potential experimental or model limitations in specific cases.

Abstract

The predictive accuracy of density functional theory (DFT) for alloy formation enthalpies is often limited by intrinsic energy resolution errors, particularly in ternary phase stability calculations. In this work, we present a machine learning (ML) approach to systematically correct these errors, improving the reliability of first-principles predictions. A neural network model has been trained to predict the discrepancy between DFT-calculated and experimentally measured enthalpies for binary and ternary alloys and compounds. The model utilizes a structured feature set comprising elemental concentrations, atomic numbers, and interaction terms to capture key chemical and structural effects. By applying supervised learning and rigorous data curation we ensure a robust and physically meaningful correction. The model is implemented as a multi-layer perceptron (MLP) regressor with three hidden layers, optimized through leave-one-out cross-validation (LOOCV) and k-fold cross-validation to prevent overfitting. We illustrate the effectiveness of this method by applying it to the Al-Ni-Pd and Al-Ni-Ti systems, which are of interest for high-temperature applications in aerospace and protective coatings.

Figures (5)

• Figure 1: A schematic illustration of the energy landscape of the heat of formation for a ternary system composed of elements (or compounds) A, B, and C. Due to the peaks and valleys in the heat of formation landscape, the phases $\alpha$, $\beta$, $\gamma$, and $\delta$ are formed.
• Figure 2: Values of $H_{\text{corr}}$ obtained from Eqn. \ref{['eq:error']} (red squares and green triangles) and from Eq.\ref{['eq:H_corr_linear']}, the linear model discussed in the text (blue dots), for all systems investigated in this study. The compounds are listed according to number of valence electrons (number given in parenthesis to the right of each chemical formula). The red squares have been used in the training set and the green triangles are used in the test set. The resulting RMSE for all the systems in the figure (both from the training set and not) is 28.7 meV/atom. Separated RMSEs for the training and prediction sets are 24.9 meV/atom and 31.4 meV/atom, respectively.
• Figure 3: Convergence of RMSE on the prediction set with the number of systems in the training set
• Figure 4: Values of $H_{\text{corr}}$ obtained from Eqn. \ref{['eq:error']} (red squares and green triangles) and from Eqn. \ref{['eq:H_corr_linear']}, the trained neural network (blue dots), for all systems investigated in this study. The compounds are listed according to number of valence electrons (number given in parenthesis to the right of each chemical formula). The red squares have been used in the training set and the green triangles are used in the test set. The resulting RMSE for all the systems in the figure (both from the training set and not) is 5.5 meV/atom. Separated RMSEs for the training and prediction sets are 2.7 meV/atom and 10.6 meV/atom, respectively.
• Figure 5: Difference between experimental heat of formation and the values obtained by correcting DFT data with the linear model (Eqn. \ref{['eq:H_pred']}, see text) using the linear model (Eqn. \ref{['eq:H_corr_linear']}, data points as red squares) and the neural network (Eqn. \ref{['eq:H_corr']}, data points as blue dots).