CAREER: Symmetries and Classical Physics in Machine Learning for Science and Engineering

物理理论的对称性描述--以及坐标自由的变换规则--在物理学的重要发展中发挥了基础性作用，包括广义相对论的发现。在现代机器学习中，对称性是深度学习架构设计的关键：从卷积神经网络的平移对称性到图神经网络的置换对称性，再到原则上是置换等变的变换器。该项目受物理学原理的启发，开发了新的数学和计算技术，以进一步利用机器学习模型设计中的对称性和微分几何。特别是，它将专注于点云和矢量场的表示学习和物理仿真。开发的技术将与纽约大学的物理学家合作应用于宇宙学和气候科学中的问题。该项目涉及约翰霍普金斯大学的博士生和巴尔的摩市公立学校的高中生实习生。该项目还包括促进拉丁美洲研究的活动，以及为数学和工程领域的妇女开展的社区建设活动。该项目的首要目标是改进表征学习技术，以自我监督的方式将文本或图像等数据嵌入潜在空间。基于最近的工作，通过近似组等方差在嵌入空间中引入了代数结构，所开发的方法将使用户能够将对输入数据的可解释修改转换为嵌入空间中的线性变换。这将通过为学习的表示提供因果结构来改进学习的嵌入的可用性。我们实现这一点隐式，使用不变的理论，并明确地，通过学习一个特殊的（解开）坐标系与微分几何技术。该项目的第二个目标是为宇宙学和气候科学开发无坐标仿真方法。一种方法是实现点云的算法，这些算法对于排列和正交（或洛伦兹）变换是不变的，可以在其上建立n体模拟。在另一种方法中，基于现代经典物理学的几何原理，为矢量和张量场构建机器学习方法，离散化到图像网格上。这些项目的成功将导致更准确的仿真与更少的昂贵的全分辨率模拟的训练集。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

The description of physical theories in terms of their symmetries –and the transformation rules of coordinate freedom– played a fundamental role in important developments in physics, including the discovery of general relativity. In modern machine learning, symmetries are key to the design of deep learning architectures: From the translation symmetry of convolutional neural networks; to the permutation symmetry of graph neural networks; to transformers, which are, in principle, permutation equivariant. This project, inspired by physics principles, develops new mathematical and computational techniques to further exploit symmetries and differential geometry in the design of machine learning models. In particular, it will focus on representation learning and physics emulation on point clouds and vector fields. The developed techniques will be applied to problems in cosmology and climate science in collaboration with physicists at New York University. The project involves PhD students from Johns Hopkins and high school student interns from Baltimore City public schools. It also includes activities to promote research in Latin America, and community-building activities for women in math and engineering.The project's first aim is to improve representation learning techniques that embed data such as text or images in a latent space in a self-supervised fashion. Based on recent work that introduced an algebraic structure in the embedding space through approximate group equivariance, the developed methods will enable users to translate interpretable modifications to the input data into linear transformations in the embedding space. This will refine the usability of the learned embeddings by providing a causal structure to the learned representations. We achieve this implicitly, using invariant theory, and explicitly, by learning a special (disentangled) coordinate system with differential geometry techniques. The project's second aim is to develop coordinate-free emulation methods for cosmology and climate science. One approach is to implement algorithms for point clouds that are invariant with respect to permutations and orthogonal (or Lorentz) transformations, on which n-body simulations can be built. In another approach, machine learning methods are built for vector and tensor fields, based on geometric principles from modern classical physics, discretized onto image grids. Success in these projects will lead to more accurate emulation with fewer expensive full-resolution simulations for the training sets.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.