Quantum element method for multi-dimensional nanostructures enabled by a projection-based learning algorithm

Abstract

A projection-based learning method developed previously based on proper orthogonal decomposition (POD), together with the quantum element method (QEM), is investigated for a 2D multi-element quantum nanostructure, where an element denotes a generic subdomain of a group of nanostructures. Unlike many other projection-based models, the basis functions for the POD approach are trained via solution data of the electron wave functions in the selected quantum state (QS) derived from direct numerical simulation of the Schrödinger equation for the nanostructure. This learning process minimizes the least square error with a small set of basis functions to reduce computational effort. Based on the QEM, the nanostructures are first partitioned into smaller generic elements (i.e., building blocks), and each of the elements is projected onto the POD space and stored in a database. For a large nanostructure, several generic elements can then be selected and glued together to perform simulation of the selected large nanostructure with the interface continuity imposed by the discontinuous Galerkin method. It has been shown that the QEM offers a reduction in numerical degrees of freedom (DoF) by 3 to 4 orders of magnitude for the trained quantum states with a high accuracy compared to direct numerical simulation. For some untrained quantum states above the trained states, a reasonably accurate prediction can be achieved with a few more DoF.