CAREER: CIF: Theory and Applications of Geometric Deep Learning

深度学习已迅速成为解决计算机视觉、语音识别和自然语言处理中大量任务的黄金标准。本质上，适当设计的人工神经网络显示了大量标记的数据点，允许其内部参数不断调整或学习。尽管这种神经网络看起来很简单，但它需要输入数据域中的某些规则才能变得有效，这就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，允许神经网络对更一般的数据领域进行操作，例如化合物或社会网络。其研究成果有可能对目前因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的广泛技术领域产生影响。除了在物理和化学领域显著提高数据驱动的发现效率的前景外，该项目中研究的方法还直接适用于天气预报或网络科学等相关学科。这个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学基础-一系列利用数据的几何性质来解决复杂任务的神经网络模型和算法-并在实际环境中展示其有效性，如物理科学。这将通过解决当前深度学习方法的两个重要限制来实现。第一个是他们学习如何以最佳计算复杂性执行算法或统计推理任务的能力，第二个是他们在缺乏图像、视频、文本或语音的规则采样结构的领域的应用。这两个目标共享几何和学习之间的基本相互作用，本项目旨在阐明这一点。这个项目追求几何稳定性的概念，这是支撑深度学习体系结构效率的数学基础，并有助于将其扩展到更一般的领域，建模为图形。这将被用来研究几何深度学习模型的优化前景和泛化误差，目前的学习理论难以解释其经验表现。最后，该项目将展示几何深度学习的有效性，并将其应用于物理科学。大多数物理系统-从原子到星系-由复杂的动力系统控制，并定义在不规则的非欧几里德域上，这对现有的深度学习体系结构提出了严重挑战。该项目旨在克服这些限制，通过将物理动力学的先验知识融入到图表结构中，建立具有自适应计算复杂性的具有自适应计算复杂性的几何深度学习模型，应用于粒子物理、化学和宇宙学。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.