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，一种是传统的，另一种是新兴的，在有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涡旋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。该项目的成果也有望使更大的科学界受益，这些科学界利用监督机器学习进行参数化模型。与此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，以及通过参与研究对本科生和研究生进行培训。高效、准确地理解系统在许多底层参数变化下的行为是普遍存在的，但由于计算成本过高，这一需求具有挑战性。有两种技术在解决这一挑战方面脱颖而出，一种是更传统的降基方法，另一种是较新的深度神经网络。本项目旨在结合这两种技术，建立一个分析驱动的计算模拟器，用于参数化偏微分方程的参数到解映射，并将所得算法应用于二维材料涡旋电子学领域。该项目的第一个主题涉及减少某些机器学习算法的泛化差距（训练期间未见数据的训练网络的性能），通过利用严格的数学方法的潜力，以本质上多层次和多分辨率的方式明智地生成计算廉价的训练数据。第二个主题的重点是对一类非线性偏微分方程的新型数值离散化及其相应的快速算法的发展，这些算法直接应用于最优质量输运。最后，该项目旨在对参数化二维材料模拟进行系统和严格的研究，包括最近发现的神奇角度扭曲双层石墨烯。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.