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：为了设计和分析在生物学、医学、材料学、电子/光子器件等领域中具有众多应用的量子纳米结构，需要进行多维度直接数值模拟（DNS）来解决薛定谔方程，由于高自由度（DoF） DNS 的庞大计算工作量可能会变得禁止。本研究利用第一原理的约简学习算法对薛定谔方程进行模拟，以实现高精度和高效率。所提出的模拟方法应用于研究两种量子点结构; 一种是在外部电场的影响下运作，另一种是在具有周期边界条件的内部势变化的影响下运作。前者类似于纳米电子器件的典型操作，后者则涉及密度泛函理论模拟和设计纳米结构和材料的应用。利用提出的方法，可以通过将 DoF 减少 3 个数量级，计算时间减少 2 个量级，与 DNS 相比，实现非常准确的预测。所提出的物理信息学习方法还能够在未经训练的量子状态下提供准确的预测，包括更高的外部场和更大的内部电势。