FRG: Collaborative Research: Singularities in Incompressible Flows: Computer Assisted Proofs and Physics-Informed Neural Networks

FRG：协作研究：不可压缩流中的奇异性：计算机辅助证明和物理信息神经网络

• 批准号: 2245021
• 负责人: Hao Jia
• 金额: $ 26.58万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2245021&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Singularities; Incompressible

Whether three-dimensional incompressible flows develop singularities in finite time and whether (weak) solutions of Navier-Stokes equations are unique, are two of the most important problems in mathematical fluid dynamics. Any progress towards resolving these problems would have significant implications for the entire field. This project integrates theoretical proofs, numerical analysis, and machine learning for understanding singularities in fluids. Our recent investigations demonstrate that intersection between mathematical proofs and deep learning offers an exciting avenue for understanding how singularity occurs in fluids. Together, the five PIs encompass strengths in several areas such as mathematical analysis, numerical simulation, or computer-assisted proofs. In addition, the project will foster collaborations and increased interactions between the researchers at several leading research universities in the US, utilizing tools developed in one field to advance another, and promote learning and training of students and postdoctoral researchers with a goal of broadening the participation of researchers from underrepresented groups in the mathematical sciences.The PIs will focus on three specific projects: (1) non-uniqueness of the Leray-Hopf solutions of the Navier Stokes equations in 3 dimensions, (2) formation of singularities for solutions of the three-dimensional Euler equations, and (3) optimization of physics-informed neural networks (PINN). Students, postdoctoral fellows, and visitors will be actively involved in these collaborations. To promote these exchanges research workshops will be organized once a year at the PIs’ institutions. These meetings will have two main objectives: a training objective, involving lectures to disseminate current ideas and progress; and an annual meeting of the PIs to review the progress and plan future steps. The PIs will also organize a summer school at Princeton University, aimed at graduate students and advanced undergraduate students. The summer school will have a scientific component, including minicourses on the mathematics of fluids, and a mentorship component, including a round table discussion regarding careers in mathematics and a women in mathematics panel.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.

三维不可压缩流是否在有限时间内产生奇点以及纳维-斯托克斯方程的（弱）解是否唯一，是数学流体动力学中最重要的两个问题，解决这些问题的任何进展都将对整个问题产生重大影响。我们最近的研究表明，数学证明和深度学习之间的交叉为理解流体中的奇点如何发生提供了一个令人兴奋的途径。五个 PI 在数学分析、数值模拟或计算机辅助证明等多个领域的优势此外，该项目还将利用在某一领域开发的工具，促进美国几所领先研究大学的研究人员之间的合作和加强互动。促进学生和博士后研究人员的学习和培训，以扩大数学科学中代表性不足群体的研究人员的参与。PI 将重点关注三个具体项目：（1）Leray-的非唯一性霍普夫解决方案三维纳维斯托克斯方程的解，(2) 三维欧拉方程解的奇点的形成，以及 (3) 物理信息神经网络 (PINN) 的优化，学生、博士后和访客将积极参与。为了促进这些交流，PI 机构每年举办一次研究研讨会，这些会议有两个主要目标：培训目标，包括传播当前想法和进展的讲座；以及年度会议。 PI 还将在普林斯顿大学组织针对研究生和高年级本科生的暑期学校，其中包括流体数学的迷你课程。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。