CAREER: CIF: Theory and Applications of Geometric Deep Learning

深度学习已经迅速成为解决计算机视觉、语音识别和自然语言处理等众多任务的黄金标准。从本质上讲，适当设计的人工神经网络显示大量标记的数据点，允许其内部参数不断调整或“学习”。尽管它看起来很简单，但这种神经网络需要在输入数据域中具有一定的规律性才能有效，这限制了它在上述领域之外的应用。例如，在自然图像中，像素被排列成规则的正方形网格，而文本和语音包含按顺序排列的样本。该项目旨在克服这一重要限制，允许神经网络在更一般的数据领域（如化合物或社会网络）上运行。其研究成果有可能影响广泛的技术领域，目前由于缺乏适当的端到端学习系统和/或计算可扩展性而受到限制。除了有望显著提高物理和化学领域数据驱动发现的效率外，本项目研究的方法还可直接应用于天气预报或网络科学等相关学科。该项目将促进物理学家、化学家、计算机科学家和应用数学家之间的跨学科合作，并将通过创建整合这些学科的新颖课程和推广活动来支持教育和多样性。该项目的目标是发展几何深度学习的数学和统计基础——一组利用数据的几何属性来解决复杂任务的神经网络模型和算法——并证明其在物理科学等实际环境中的有效性。这将通过解决当前深度学习方法的两个重要限制来实现。首先是它们学习如何以最佳计算复杂度执行算法或统计推理任务的能力，其次是它们在缺乏图像、视频、文本或语音的常规采样结构的领域中的应用。这两个目标共享几何和学习之间的基本相互作用，该项目旨在阐明。该项目追求几何稳定性的概念，这是支撑深度学习架构效率的数学基础，并促进其扩展到更一般的领域，以图为模型。这将用于研究几何深度学习模型的优化景观和泛化误差，目前的学习理论正在努力解释其经验性能。最后，该项目将展示几何深度学习在物理科学应用中的有效性。大多数物理系统——从原子到星系——都是由复杂的动力系统控制的，并且是在不规则的非欧几里得域上定义的，这对现有的深度学习架构提出了严峻的挑战。该项目旨在通过构建具有自适应计算复杂性的“物理感知”几何深度学习模型来克服这些限制，该模型应用于粒子物理学、化学和宇宙学，通过将物理动力学的先验知识整合到图结构中。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。

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.