LEAPS-MPS: Deep Learning the Knot Landscape

该奖项由《2021年美国救援计划法案》(公法117-2)提供全部资金。纽结理论的领域始于19世纪中期，主要是受到物理学思想的推动。今天，纽结通过规范理论和量子场论在物理学中扮演着重要的角色，通过蛋白质和DNA纽结的分子生物学，以及以4-流形的手体图和3-流形的Dehn手术描述形式的低维拓扑。2016年，PI开创了一种应用机器学习和人工智能技术研究结点理论问题的新方法。虽然现在在这个方向上有一小部分但不断增长的研究，来自数学家、物理学家和计算机科学家的贡献，但现有的大部分工作集中在监督学习技术和强化学习在解开辫子方面的应用。PI将采用产生式机器学习和强化学习的新技术来研究节点的拓扑性质，学习节点及其不变量的潜在分布。PI还将通过建立新观测和实验结果的理论基础，扩展与合作者的早期工作。作为该项目的一部分，PI将为本科生开发一个有指导的数据科学研究培训计划，在该计划中，学生将由PI和一名来自行业或学术界的数据科学家指导。学生们将与PI一起参加一个为期一学期的指导学习小组，然后完成一个密集的研讨会，在那里他们与外部导师一起解决行业中的数据科学问题。在这一计划的每一步，都将特别注意通过与专门接触这些社区的校园组织建立伙伴关系，增加历史上代表性不足的群体的参与。通过参与辅导性研究，这些学生将获得帮助他们准备在学术界和工业界获得研究生学位和职业的经验，从而为来自这些代表性不足的社区的学生未来的榜样做准备。PI将适应文本到图像生成的对抗性网络来构建不变到节点的GAN，允许构建具有指定拓扑属性的节点。PI还将使用变分自动编码器来学习新的潜在节点分布，这些节点相对于各种拓扑属性和不变量是自然的。这些节点的潜在表示将提供对节点分布的更清楚的理解，并产生随机模型，从而允许有针对性地生成具有特定属性的节点。节点理论数据的任何新的潜在表示将被其他研究人员用于训练新的机器学习模型，从而提高正在开发的模型的性能。这些技术将被用于指导对重要的公开猜想进行反例搜索。此外，PI将使用深度强化学习算法来研究切片亏格和编织带排序问题，将现有关于使用强化学习的结果推广到编织的解开。考虑到计算纽结的切片亏格的问题是低维拓扑和物理结构中关键开放问题的核心，这里开发的新技术将在纽结理论之外有直接的应用。这些技术的成功使用将成为未来生成式机器学习和强化学习应用于其他数学领域的模板。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。

This award is funded in whole under the American Rescue Plan Act of 2021 (Public Law 117-2). The field of knot theory began in the mid 1800s, motivated heavily by ideas from physics. Today, knots play an important role in physics through gauge theory and quantum field theories, molecular biology via protein and DNA knotting, and low-dimensional topology in the form of handlebody diagrams of 4-manifolds and Dehn surgery descriptions of 3-manifolds. In 2016 the PI initiated a novel approach to studying problems in knot theory by applying techniques of machine learning and artificial intelligence. While there is now a small but growing body of research in this direction, with contributions from mathematicians, physicists, and computer scientists, most of the existing work focuses on techniques of supervised learning and applications of reinforcement learning to unknotting braids. The PI will adapt new techniques from generative machine learning and reinforcement learning to study topological properties of knots and learn latent distributions of knots and their invariants. The PI will also extend earlier work with collaborators, by establishing theoretical underpinnings of new observations and experimental results. As part of this project the PI will develop a mentored data science research training program for undergraduate students, in which students will be mentored by both the PI and a data scientist from industry or academia. Students will participate in a semester-long mentored learning group with the PI, before completing an intensive workshop where they work together to solve data science problems from industry with their external mentor. At each step of this program, special attention will be given to increasing participation of historically underrepresented groups through partnerships with campus organizations that specialize in outreach to these communities. By participating in mentored research these students will gain experience that will help them prepare for graduate degrees and careers in academia and industry, thereby preparing to be future role models for students from these underrepresented communities.The PI will adapt text-to-image generative adversarial networks to construct invariant-to-knot GANs, allowing for the construction of knots with prescribed topological properties. The PI will also use variational autoencoders to learn new latent distributions of knots which are natural with respect to various topological properties and invariants. These latent representations of knots will provide a clearer understanding of knot distributions and produce random models that allow for targeted generation of knots with specified properties. Any new latent representations of knot theoretic data will be made available to other researchers for use in training new machine learning models, improving the performance of the models being developed. These techniques will be used to guide searches for counterexamples to important open conjectures. In addition, the PI will use deep reinforcement learning algorithms to study the slice genus and braid band rank problems, generalizing existing results on the use of reinforcement learning to the unknotting of braids. Given that the problem of computing the slice genus of knots is central to key open questions in low-dimensional topology and constructions in physics, new techniques developed here will have direct applications outside of knot theory. Successful use of these techniques will serve as a template for future applications of generative machine learning and reinforcement learning to other areas of mathematics.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.