Multi-dimensional direct numerical simulation (DNS) of the Schr\"odinger equation is needed for design and analysis of quantum nanostructures that offer numerous applications in biology, medicine, materials, electronic/photonic devices, etc. In large-scale nanostructures, extensive computational effort needed in DNS may become prohibitive due to the high degrees of freedom (DoF). This study employs a reduced-order learning algorithm, enabled by the first principles, for simulation of the Schr\"odinger equation to achieve high accuracy and efficiency. The proposed simulation methodology is applied to investigate two quantum-dot structures; one operates under external electric field, and the other is influenced by internal potential variation with periodic boundary conditions. The former is similar to typical operations of nanoelectronic devices, and the latter is of interest to simulation and design of nanostructures and materials, such as applications of density functional theory. Using the proposed methodology, a very accurate prediction can be realized with a reduction in the DoF by more than 3 orders of magnitude and in the computational time by 2 orders, compared to DNS. The proposed physics-informed learning methodology is also able to offer an accurate prediction beyond the training conditions, including higher external field and larger internal potential in untrained quantum states.

翻译：多维度薛定谔方程的直接数值模拟是设计和分析量子纳米结构所必需的，这类结构在生物学、医学、材料学、电子/光子器件等领域具有广泛应用。在大尺度纳米结构中，直接数值模拟所需的巨大计算量可能因自由度过高而变得不可行。本研究采用了一种基于第一性原理的降阶学习算法来模拟薛定谔方程，以实现高精度和高效率。所提出的模拟方法被用于研究两种量子点结构：一种在外部电场作用下运行，另一种受周期边界条件下的内部势场变化影响。前者类似于纳米电子器件的典型工作状态，后者则对纳米结构与材料的模拟与设计（如密度泛函理论的应用）具有重要意义。采用本方法后，与直接数值模拟相比，自由度降低了三个数量级以上，计算时间降低了两个数量级，同时实现了非常精确的预测。所提出的物理信息学习方法还能够在训练条件之外提供准确预测，包括未训练量子态下的更高外部电场和更大内部势场。