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.

翻译：采用扩展Hubbard泛函的密度泛函理论（DFT+$U$+$V$）为精确描述含过渡金属或稀土元素的复杂材料提供了稳健框架。该框架通过缓解半局域泛函固有的自相互作用误差实现这一目标，此类误差在具有部分填充$d$和$f$电子态的体系中尤为显著。然而，该方法的精确性取决于在位$U$与位间$V$ Hubbard参数的准确确定。实践中，这些参数通常通过需要先验知识的半经验调参获得，或更严谨地采用预测性强但计算昂贵的第一性原理计算获取。本文提出一种基于等变神经网络的机器学习模型，该模型以原子占据矩阵作为描述符，直接捕获目标体系的电子结构、局域化学环境及氧化态。我们重点预测通过迭代线性响应计算（按密度泛函微扰理论（DFPT）实现）与结构弛豫自洽计算的Hubbard参数。值得注意的是，当使用涵盖不同晶体结构与成分的11种材料数据进行训练时，我们的模型对Hubbard $U$和$V$参数分别实现了3%和5%的平均相对绝对误差。通过规避计算昂贵的DFT或DFPT自洽流程，该模型以可忽略的计算开销显著加速Hubbard参数预测，同时逼近DFPT的精度。此外，凭借其强大的可迁移性，该模型可通过高通量计算加速材料发现与设计进程，对多种技术应用具有重要意义。