Learning High-Dimensional Non-Linear Maps Arising from Physical Phenomena via Symmetry and Structure-Preserving Deep Neural Networks

近年来，机器学习，特别是深度学习，推动了语言和图像处理领域的几项突破。在其在商业和医学应用中取得成功之后，已经做出了一些努力，将深度学习应用范围扩展到科学和工程任务。这一领域的进展受到两个主要问题的阻碍：科学任务往往有更严格的准确性要求，所需数据通常要么稀缺，要么获取成本高昂。然而，其中一些任务具有大量的相关知识，通过整合特定于问题的知识，新的深度学习架构可以实现必要的准确性，同时需要更少的数据。该项目的目标有两个：1）设计深度学习系统，绕过计算昂贵的非线性系统的经典模拟，应用于量子化学和太阳能电池和纳米材料等新材料的设计; 2）扩展经典非线性方法的能力，例如通过引入新型神经网络为生物医学，雷达和地震成像产生更清晰的图像。该项目将为学生提供跨学科的应用数学培训和研究经验。该项目的目标是设计专门为两个科学应用量身定制的深度神经网络：密度泛函理论，用于在微观水平上模拟材料，以及逆问题，其专注于从边界数据恢复感兴趣的量，例如在CT，MRI，和超声成像。该项目的主要成果将是利用每个应用程序的物理和分析特性来满足基础物理的对称性和结构的新型网络。对于第一个应用程序，目标是从原子核的位置创建给定系统的电子密度的有效表示，从而绕过密度泛函理论框架内的Kohn-Sham方程的解。在第二个应用程序中，考虑由亥姆霍兹方程建模的逆散射问题。该项目的重点是受蝴蝶算法启发的网络，蝴蝶算法是一种快速方法，用于处理用于描述线性逆散射问题的傅立叶积分算子。一个非线性推广的蝴蝶算法，被称为蝴蝶网，将被扩展到处理数据在几个频率以下的二元分区，以稳定的训练步骤。由此产生的宽带蝴蝶网络将用于研究低于奈奎斯特水平的细节的超分辨率。该奖项反映了NSF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。

Machine learning, in particular deep learning, has fueled several breakthroughs in language and image processing in recent years. Following its success in applications to business and medicine, several efforts have been made to extend the range of deep learning applications to scientific and engineering tasks. Progress in this area has been hampered by two main issues: scientific tasks often have more stringent accuracy requirements, and the required data is usually either scarce or expensive to obtain. However, some of these tasks have vast pools of associated knowledge, and by incorporating problem-specific knowledge, new deep learning architectures can achieve the necessary accuracy while requiring significantly less data. The objective of this project is twofold: 1) to design deep learning systems that bypass computationally-expensive classical simulations of nonlinear systems, with applications in quantum chemistry and design of new materials such as solar cells and nano-materials; and 2) to expand the capabilities of classical nonlinear methods, for example by introducing novel neural networks to produce sharper images for biomedical, radar, and seismic imaging. This project will provide interdisciplinary applied mathematics training and research experiences for students.The goal of this project is to design deep neural networks specifically tailored for two scientific applications: density functional theory, which is used to simulate materials at the microscopic level, and inverse problems, which are focused on recovering quantities of interest from boundary data, such as is done in CT, MRI, and ultrasound imaging. The main result of the project will be novel networks that leverage the physical and analytical properties of each application to satisfy the symmetries and structures of the underlying physics. For the first application, the goal is to create an efficient representation of the electron density of a given system from the position of the nuclei, thus bypassing the solution of the Kohn-Sham equations within the density functional theory framework. In the second application, inverse scattering problems modeled by the Helmholtz equation are considered. The project focuses on networks inspired by the butterfly algorithm, a fast method tailored to handle Fourier integral operators that are used to describe the linearized inverse scattering problem. A nonlinear generalization of the butterfly algorithm, dubbed a butterfly-net, will be extended to handle data at several frequencies following a dyadic partition, to stabilize the training step. The resulting wide-band butterfly network will be used to study the super-resolution of details below the Nyquist level.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.