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：量子纳米结构在生物学、医学、材料学、电子/光子器件等领域具有广泛应用，其设计与分析需要对薛定谔方程进行多维直接数值模拟(DNS)。在大尺度纳米结构中，由于高自由度(DoF)的存在，DNS所需的庞大计算量可能变得难以承受。本研究采用一种基于第一性原理的降阶学习算法，用于薛定谔方程的高精度、高效率模拟。所提出的模拟方法被应用于研究两种量子点结构：一种在外加电场下工作，另一种受周期性边界条件下内部电势变化的影响。前者类似于纳米电子器件的典型工作模式，后者则与纳米结构和材料的模拟与设计相关，如密度泛函理论的应用。采用所提方法，相较于DNS，可实现自由度降低超过3个数量级、计算时间降低2个数量级的极高精度预测。该物理信息学习方法还能在训练条件之外实现准确预测，包括未训练量子态下的更高外加电场和更大内部电势。