CAREER: CIF: Theory and Applications of Geometric Deep Learning

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Deep Learning has quickly become the gold standard for solving a multitude of tasks in computer vision, speech recognition, and natural language processing. In essence, appropriately designed artificial neural networks are shown large quantities of labeled data-points, allowing their internal parameters to be continually adjusted or 'learned'. Despite its apparent simplicity, such neural networks require certain regularities in the input data domain in order to become effective, which limits its application to areas beyond those described above. For example, in natural images, pixels are arranged into regular squared grids, whereas text and speech contains samples sequentially aligned. This project aims at overcoming this important limitation, by allowing neural networks to operate on far more general data domains, such as chemical compounds or social networks. Its research outcomes have the potential to impact a broad range of technological areas currently limited by lack of appropriate end-to-end learning systems and/or computational scalability. Besides the prospect of significantly increasing the efficiency of data-driven discoveries in physics and chemistry domains, the methods studied in this project directly apply to related disciplines such as weather forecasting or network science. This project will foster cross-disciplinary collaborations between physicists, chemists, computer scientists and applied mathematicians, and will support education and diversity by creating novel courses and outreach activities integrating these disciplines.The goal of this project is to develop the mathematical and statistical foundations of geometric deep learning---a family of neural network models and algorithms that leverage geometric properties of the data in order to solve complex tasks---and to demonstrate its effectiveness in practical settings such as the physical sciences. This will be enabled by addressing two important limitations of current deep learning methods. The first is their ability to learn how to perform algorithmic or statistical inference tasks with optimum computational complexity, which the second is their application to domains that lack the regular sampling structure of images, video, text or speech. Both objectives share a fundamental interplay between geometry and learning that this project aims to elucidate. This project pursues the notion of geometric stability, the mathematical foundation that underpins the efficiency of deep learning architectures and facilitates its extension to more general domains, modeled as graphs. This will be used to study the optimization landscape and generalization error of geometric deep learning models, where current learning theory struggles to explain its empirical performance. Finally, the project will demonstrate the effectiveness of geometric deep learning with applications to physical sciences. Most physical systems---from atoms to galaxies---are governed by complex dynamical systems and are defined over irregular, non-Euclidean domains, presenting a serious challenge for existing deep learning architectures. This project seeks to overcome these limitations by building 'physics-aware' geometric deep learning models with adaptive computational complexity, applied to particle physics, chemistry and cosmology, by incorporating prior knowledge of the physical dynamics into the graph structure.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.