Reduced Basis Enhancements of Neural Networks and Their Application to Quantum Materials Simulation

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This project combines two types of numerical techniques, one traditional and the other nascent, in the context of efficient multi-parametric simulations. The project aims to develop a new algorithm based on deep neural networks that will be applied to the simulation of quantum materials in the field of two-dimensional materials twistronics toward a fast and accurate configuration-to-performance map. Outcomes of this project are expected also to benefit the greater scientific community that utilizes supervised machine learning for parameterized models. Software associated with this project will be made freely available. The project involves development of graduate coursework and the training of undergraduate and graduate students through involvement in the research. The need to understand the behavior of a system efficiently and accurately under variation of many underlying parameters is ubiquitous yet challenging due to the prohibitively high computational cost. Two techniques stand out in addressing this challenge, the more traditional reduced basis method and the newer deep neural networks. This project aims to combine these two techniques to build an analysis-driven computational emulator for the parameter-to-solution map of parameterized partial differential equations and to apply the resulting algorithm to the field of 2D materials twistronics. The first theme of the project involves reducing the generalization gap (performance of the trained network on data unseen during training) of certain machine learning algorithms by tapping the potential of a rigorous mathematical approach in judiciously generating computationally cheap training data in an intrinsically multi-level and multi-resolution fashion. The focus of the second theme is the development of novel numerical discretizations and their corresponding fast algorithms for a class of nonlinear partial differential equations that has direct application in optimal mass transport. Finally, the project aims to provide a systematic and rigorous study of parameterized 2D materials simulation, including the recently discovered magical angle twisted bilayer graphene.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.