Quantum nanostructures, including quantum wells, wires and dots, have been used for decades to improve performance of electronic and photonic devices, such as nanoscale transistors, lasers, light emitting diodes (LEDs), photovoltaics, etc. To understand device characteristics for design of nanoscale electronic and photonic devices, electron wave functions (WFs) with their eigenenergies need to be solved from the Schr¨odinger equation subjected to internal/external electric field or potential. For accurate analysis of electron WFs, numerical simulations of the Schr¨odinger equation for nanostructures are always needed, which is however computational intensive, especially for 3D quantum structures. Efficiency and accuracy of multidimensional simulations of quantum eigenvalue problems for these nanoscale structures are thus crucial for cost-effective design of these structures and for further advances in these modern technologies. To offer an effective approach for solving the Schr¨odinger equation, a physics-based learning algorithm enabled by proper orthogonal decomposition (POD) has been proposed. Via the POD, the Schr¨odinger equation is projected onto the eigenspace (or POD space) constituted by a finite set of basis functions (or POD modes). These modes are trained by WF data collection from the direct numerical simulations (DNSs) of the Schr¨odinger equation. To minimize the training effort for large nanostructures and to utilize the building block concept for cost-effective design, the structures are partitioned into generic elements, and each of the selected elements is trained and projected onto its POD modes. These trained generic elements are then stored in a database and can then then be selected and glued together to construct a large domain for cost-effective simulation and design of nanostructure. Quantum dot structures will be investigated to demonstrate the effectiveness of the multi-element quantum POD simulation methodology.