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

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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.