Machine learning Hubbard parameters with equivariant neural networks

TL;DR

The paper presents an equivariant neural-network approach to predict self-consistent Hubbard parameters $U$ and $V$ in DFT+$U$+$V$ calculations directly from atomic-occupation descriptors. By encoding local electronic structure through on-site occupation matrices and enforcing SE(3) symmetry via irreducible representations, the model achieves high accuracy (about 3% for $U$ and 5% for $V$) across a diverse set of materials using a relatively small training set. It demonstrates strong robustness to reduced iteration data, and notable transferability across oxidation states and crystal structures, enabling rapid, high-throughput screening with negligible computational overhead compared to full first-principles protocols. These results suggest a practical pathway to accelerate materials discovery in systems with localized d/f electrons while preserving accuracy close to DFPT benchmarks.

Abstract

Density-functional theory with extended Hubbard functionals (DFT+$U$+$V$) provides a robust framework to accurately describe complex materials containing transition-metal or rare-earth elements. It does so by mitigating self-interaction errors inherent to semi-local functionals which are particularly pronounced in systems with partially-filled d and f electronic states. However, achieving accuracy in this approach hinges upon the accurate determination of the on-site $U$ and inter-site $V$ Hubbard parameters. In practice, these are obtained either by semi-empirical tuning, requiring prior knowledge, or, more correctly, by using predictive but expensive first-principles calculations. Here, we present a machine learning model based on equivariant neural networks which uses atomic occupation matrices as descriptors, directly capturing the electronic structure, local chemical environment, and oxidation states of the system at hand. We target here the prediction of Hubbard parameters computed self-consistently with iterative linear-response calculations, as implemented in density-functional perturbation theory (DFPT), and structural relaxations. Remarkably, when trained on data from 12 materials spanning various crystal structures and compositions, our model achieves mean absolute relative errors of 3% and 5% for Hubbard $U$ and $V$ parameters, respectively. By circumventing computationally expensive DFT or DFPT self-consistent protocols, our model significantly expedites the prediction of Hubbard parameters with negligible computational overhead, while approaching the accuracy of DFPT. Moreover, owing to its robust transferability, the model facilitates accelerated materials discovery and design via high-throughput calculations, with relevance for various technological applications.

Figures (5)

• Figure 1: Calculating Hubbard corrections self-consistently using density-functional perturbation theory. a) Protocol for the self-consistent calculation of Hubbard parameters using . Timrov:2021$U_\mathrm{in}$ and $V_\mathrm{in}$ represent the input Hubbard parameters, while $U_\mathrm{out}$ and $V_\mathrm{out}$ denote the output parameters, with $\Delta$ representing the convergence threshold. $U_\mathrm{SC}$ and $V_\mathrm{SC}$ are the final Hubbard parameters. b) Convergence of the Hubbard $U$ parameter for Mn-3d states in LiMnPO$_4$ using the self-consistent protocol.Timrov:2022b The inset displays the crystal structure of the material, where arrows indicate the spin direction, and Li atoms are depicted in grey, O in red, Mn in violet, and P in yellow.
• Figure 2: Schematic illustration of the equivariant neural-network ML model for predicting $U$ and $V$ Hubbard parameters.$U_\mathrm{in}$ and $V_\mathrm{in}$ are the input Hubbard parameters, while $U_\mathrm{out}$ and $V_\mathrm{out}$ are the outputs. The atomic species enter as one-hot tensors, and $r_{IJ}$ is the interatomic distance between sites $I$ and $J$. $\mathbf{x}^1_\mathrm{d}$ and $\mathbf{x}^2_\mathrm{d}$ are the occupation matrices for the d orbitals, while $\mathbf{x}^1_\mathrm{p}$ and $\mathbf{x}^2_\mathrm{p}$ for the p orbitals, all in the special representation for the ML model [see \ref{['eq:occ_1', 'eq:occ_2']}]. Tensors are represented as squares: open for inputs and outputs, and filled for intermediate features.
• Figure 3: Parity plots showing the prediction accuracy on an unseen validation dataset. The energies in the legend are the RMSE categorized by element(s) and the overall RMSE across all elements. All attributes listed in \ref{['tab:representations']} are used as inputs for the ML model.
• Figure 4: Evaluation of model performance using a reduced number of self-consistent steps. RMSE for all materials as a function of the number of iterations $N_\mathrm{iter}$ in the protocol (see \ref{['fig:self-consistent']}). At each $N_\mathrm{iter}$ the plotted reference value is the between the result obtained from the previous iteration and the current one (i.e. a measure of how much the Hubbard $U$ changes by doing one more iteration). The value shows the prediction made from having trained on results from all the preceding $N_\mathrm{iter} - 1$ iterations and predicting the $N_\mathrm{iter}$th value. This helps to answer the question of how much the model can improve the Hubbard parameter, at essentially no cost, when limiting the number of steps as might be done during the high-throughput screening.
• Figure 5: Evaluation of model transferability to unseen electronic structures. RMSE as a function of $N_\mathrm{c}$ (the number of Li concentrations) for three olivines. The model is trained using $N_\mathrm{c}$ concentrations and validated on the remaining $5-N_\mathrm{c}$. Error bars indicate the range of RMSE values obtained by considering various permutations of concentrations of the training and validation datasets, large dots represent the average, small dots show individual results and experiments that only contain training data from a single oxidation state are labelled ($2+$ and $3+$). The reference is the RMSE computed by training the ML model on 80% of the Hubbard parameters from all five concentrations, with validation performed on the remaining 20% after de-duplication. The blue line represents the mean results over three runs with different random initializations, the confidence interval shows the standard deviation.