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: 2245228
• 负责人: Alexandru Ionescu
• 金额: $ 60.81万
• 依托单位: Princeton University
• 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=2245228&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. Recent investigations by the PIs 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.

三维不可压缩流在有限时间内是否出现奇点以及Navier-Stokes方程的（弱）解是否唯一，是数学流体力学中两个最重要的问题。解决这些问题的任何进展都将对整个领域产生重大影响。该项目集成了理论证明，数值分析和机器学习，以理解流体中的奇异性。PI最近的调查表明，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条令人兴奋的途径。总之，五个PI包括几个领域的优势，如数学分析，数值模拟，或计算机辅助证明。此外，该项目还将促进美国几所领先研究型大学的研究人员之间的合作和互动，利用一个领域开发的工具推动另一个领域的发展，并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的群体的研究人员的参与。（1）三维Navier Stokes方程的Leray-Hopf解的非唯一性，（2）三维Euler方程解的奇点形成，以及（3）物理信息神经网络（PINN）的优化。学生，博士后研究员和访问者将积极参与这些合作。为了促进这些交流，将每年在公共机构举办一次研究讲习班。这些会议将有两个主要目标：一个是培训目标，包括举办讲座，传播当前的想法和进展;另一个是主持人年度会议，审查进展情况，规划今后的步骤。PI还将在普林斯顿大学组织一个针对研究生和高级本科生的暑期学校。暑期学校将有一个科学的组成部分，包括关于流体数学的迷你课程，和一个导师组成部分，包括一个关于数学职业的圆桌讨论会和一个数学界的妇女小组。这个奖项反映了NSF的法定使命，并被认为是值得支持的，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。