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.

三维不可压缩的流是否在有限的时间内发展出奇异性，以及Navier-Stokes方程的（弱）解决方案是否是唯一的，这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。该项目集成了理论证明，数值分析和机器学习，以了解流体中的奇异性。 PIS的最新调查表明，数学证据与深度学习之间的交集为了解烟道中的奇异性是一种令人兴奋的途径。这五个PI共同增强了多个领域的优势，例如数学分析，数值模拟或计算机辅助证明。此外，该项目将促进美国几所领先研究大学的研究人员之间的合作和增加的互动，使用一个领域开发的工具来推进另一个领域，并促进对学生和博士后研究人员的学习和培训，目的是扩大代表性不足的小组的研究人员参与数学科学的研究人员的参与。三个维度的方程式，（2）三维EULER方程溶液的奇异性形成，以及（3）优化物理知识的神经网络（PINN）。学生，博士后研究员和游客将积极参与这些合作。为了促进这些交流，研究研讨会将每年在PIS机构中进行一次。这些会议将有两个主要目标：培训目标，涉及讲座以传播当前的思想和进步；以及PI的年度会议，以审查进度并计划未来的步骤。 PIS还将在普林斯顿大学组织一所暑期学校，针对研究生和高级本科生。暑期学校将有一个科学的组成部分，包括有关流体数学的微型护理以及心态的组成部分，包括有关数学职业的圆桌讨论和数学小组中女性的圆桌讨论。该奖项反映了NSF的法定任务，并通过使用该基金会的知识优点和广泛影响来评估NSF的法定任务。