Physics-aware POD-based learning for Ab initio QEM-Galerkin simulations of periodic nanostructures

Quantum nanostructures offer crucial applications in electronics, photonics, materials, drugs, etc. For accurate design and analysis of nanostructures and materials, simulations of the Schrӧdinger or Schrӧdinger-like equation are always needed. For large nanostructures, these eigenvalue problems can be computationally intensive. One effective solution is a learning method via Proper Orthogonal Decomposition (POD), together with ab initio Galerkin projection of the Schrӧdinger equation. POD-Galerkin projects the problem onto a reduced-order space with the POD basis representing electron wave functions (WFs) guided by the first principles in simulations. To minimize training effort and enhance robustness of POD-Galerkin in larger structures, the quantum element method (QEM) was proposed previously, which partitions nanostructures into generic quantum elements. Larger nanostructures can then be constructed by the trained generic quantum elements, each of which is represented by its POD-Galerkin model. This work investigates QEM-Galerkin thoroughly in multi-element quantum-dot (QD) structures on approaches to further improve training effectiveness and simulation accuracy and efficiency for QEM-Galerkin. To further improve computing speed, POD and Fourier bases for periodic potentials are also examined in QEM-Galerkin simulations. Results indicate that, considering efficiency and accuracy, the POD potential basis is superior to the Fourier potential basis even for periodic potentials. Overall, QEM-Galerkin offers more than a 2-order speedup in computation over direct numerical simulation for multi-element QD structures, and more improvement is observed in a structure comprising more elements.