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

三维不可压缩流动是否在有限时间内出现奇性，以及Navier-Stokes方程的(弱)解是否唯一，是数学流体力学中的两个重要问题。在解决这些问题方面取得的任何进展都将对整个领域产生重大影响。这个项目集成了理论证明、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的交叉为理解流体中的奇点是如何发生的提供了一条令人兴奋的途径。总而言之，这五项绩效指标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加强互动，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，目标是扩大来自未被充分代表的群体的研究人员在数学科学中的参与。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、博士后研究员和来访者将积极参与这些合作。为促进这些交流，将每年在私人投资机构举办一次研究讲习班。这些会议将有两个主要目标：一是培训目标，包括举办讲座以传播当前的想法和进展；以及一次私人投资机构年度会议，以审查进展情况并规划未来的步骤。PIS还将在普林斯顿大学组织一个暑期班，针对研究生和高级本科生。暑期班将包括科学部分，包括流体数学的微型课程，以及指导部分，包括关于数学职业的圆桌讨论和数学中的女性小组讨论。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。

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.