AI 中文总结
研究旨在填补电子结构计算方法空白，核心方法是用E(3)等变图神经网络学习电子结构矩阵，主要贡献是构建模块化框架Mandala，支持多种功能，能评估可观测量，可补充原子间势工作流程。
AI 中文摘要
基于Kohn-Sham密度泛函理论的电子结构计算在计算材料科学和化学中不可或缺，但其计算成本限制了可处理的系统规模和模拟时间。传统机器学习原子间势通常只针对能量和力，遗漏了重建能带结构、态密度、空间电荷分布等所需的量子算符级信息。Mandala填补了这一方法空白，它是一个用于用E(3)等变图神经网络学习块稀疏电子结构矩阵的模块化软件框架。该框架围绕原子分辨哈密顿量、重叠和密度矩阵的统一表示构建，支持多种功能。Mandala可直接从预测算符评估选定的可观测量，将电子结构学习与可观测量指导建模联系起来，旨在通过在一个可扩展实现中解析电子结构和算符衍生的可观测量来补充原子间势工作流程。
英文摘要
Electronic-structure calculations based on Kohn-Sham density functional theory remain indispensable in computational materials science and chemistry. Their computational cost, however, limits accessible system sizes and simulation times. At the same time, conventional machine-learning interatomic potentials (MLIPs), which are becoming the workhorse of large-scale materials modeling, usually target only energies and forces. They therefore leave out the quantum-operator-level information required to reconstruct band structures, densities of states, spatial charge distributions, and other electronic observables. \texttt{Mandala} fills this methodological gap. It is a modular software framework for learning block-sparse electronic-structure matrices with E(3)-equivariant graph neural networks. The framework is built around a unified representation of atom-resolved Hamiltonian, overlap, and density matrices, together with reusable abstractions for basis conversion, sparse block handling, irreducible representation mapping, graph construction, model definition, and training. This design allows \texttt{Mandala} to support heterogeneous chemical compositions, a wide range of neural architecture variants within one workflow, and multiple electronic-structure backends. \texttt{Mandala} evaluates selected observables directly from the predicted operators, including band energy, electron count, density of states, and band structure. This connects electronic-structure learning and observable-guided modeling while retaining a representation tied to quantum-mechanical operators rather than only scalar or vector targets as in MLIPs. In this form, \texttt{Mandala} is intended to complement atomistic interatomic potential workflows by resolving electronic structure and operator-derived observables within one scalable implementation.
