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: 2245017
• 负责人: Javier Gomez Serrano
• 金额: $ 32.52万
• 依托单位: Brown 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=2245017&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.

三维不可压缩流动是否在有限时间内出现奇点以及Navier-Stokes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领域产生重大影响。这个项目整合了理论证明、数值分析和机器学习来理解流体中的奇点。我们最近的研究表明，数学证明和深度学习之间的交集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，这五个pi涵盖了几个领域的优势，如数学分析、数值模拟或计算机辅助证明。此外，该项目将促进美国几所顶尖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来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，以扩大数学科学领域代表性不足群体的研究人员的参与。pi将专注于三个具体项目：(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，博士后研究员和参观者将积极参与这些合作。为了促进这些交流，每年将在pi的机构组织一次研究研讨会。这些会议将有两个主要目标：培训目标，包括传播当前思想和进展的讲座；以及召开年度pi会议，审查进展情况并规划未来步骤。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部分，包括流体数学的迷你课程，以及一个指导部分，包括关于数学职业的圆桌讨论和一个数学女性小组。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。