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 11 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.

用具有扩展 Hubbard 泛函的密度泛函理论（DFT + U + V）来精确描述含有过渡金属或稀土元素的复杂材料，并基于等变神经网络的机器学习模型，通过捕捉系统的电子结构、局部化学环境和氧化态来预测 Hubbard 参数，从而加速材料的发现和设计。