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

该项目结合了两种类型的数值技术，一个传统的和其他新生的，在高效的多参数模拟的背景下。该项目旨在开发一种基于深度神经网络的新算法，该算法将应用于二维材料双电子学领域的量子材料模拟，以实现快速准确的配置到性能映射。该项目的成果预计也将有利于更大的科学界，利用监督机器学习参数化模型。与该项目相关的软件将免费提供。该项目涉及研究生课程的开发和通过参与研究对本科生和研究生进行培训。在许多基本参数变化的情况下，需要有效和准确地理解系统的行为是普遍存在的，但由于过高的计算成本而具有挑战性。在应对这一挑战方面，有两种技术脱颖而出，即更传统的缩减基方法和更新的深度神经网络。本计画旨在将这两种技术联合收割机结合起来，建立一个分析驱动的参数化偏微分方程参数-解映射的计算仿真器，并将所得的算法应用于二维材料双电子学领域。该项目的第一个主题涉及通过挖掘严格的数学方法的潜力来减少某些机器学习算法的泛化差距（训练网络在训练期间看不见的数据上的性能），以本质上多层次和多分辨率的方式明智地生成计算成本低的训练数据。第二个主题的重点是发展新的数值离散和相应的快速算法的一类非线性偏微分方程，有直接的应用在最佳质量传输。最后，该项目旨在对参数化二维材料模拟进行系统而严格的研究，包括最近发现的神奇角度扭曲双层石墨烯。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.