Computer-assisted Proofs in Fluid Mechanics and Applications

流体力学及其应用中的计算机辅助证明

• 批准号: 2247537
• 负责人: Javier Gomez Serrano
• 金额: $ 22.63万
• 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=2247537&HistoricalAwards=false
• 关键词: Computer; assisted; Proofs; Fluid; Mechanics

The research goal of this project is twofold: on the one hand to pursue methods and ideas developed in recent work in the search for either singularities or global existence in incompressible fluids with finite energy and on the other to transfer the techniques to solve long-standing open problems in spectral geometry and mathematical physics. A key ingredient in its success is to have accurate numerics, together with a deep understanding of the regularity theory. Therefore, the interdisciplinary nature of the project, which involves numerical computations, physics informed neural networks (PINN), computer-assisted proofs, modern methods in partial differential equations (PDE) and harmonic analysis, is an essential ingredient for its successful outcome. The recent emergence of techniques blending mathematical analysis and deep learning offers a new opportunity for developing new angles of attack and better understanding classical problems in mathematics. Undergraduate and graduate students, postdoctoral researchers, and research visitors actively participate in the work of this project. In particular, undergraduate students from underrepresented groups in the mathematical sciences are mentored and involved in research experiences integral to the project, on problems that are rich and suitable for learning and partial progress, both numerically and analytically. The project is structured in three blocks, the first one involving the incompressible Euler equations, as well as other water wave models; the second on the surface quasi-geostrophic (SQG), the generalized-SQG equations, and related models; and a third one on applications to spectral geometry and mathematical physics.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.

该项目的研究目标是双重的：一方面，在搜索有限能量的不可压缩液体中，在寻找奇异性或全球存在的方法中提出的方法和思想，另一方面是在光谱几何学和数学物理学中解决长期存在的开放问题。成功的关键要素是具有准确的数字以及对规则性理论的深刻理解。因此，该项目的跨学科性质涉及数值计算，物理知情的神经网络（PINN），计算机辅助的证明，偏微分方程中的现代方法（PDE）和谐波分析，是其成功结果的重要组成部分。融合数学分析和深度学习的技术的最新出现为发展新的攻击角度并更好地理解数学中的经典问题提供了新的机会。本科生和研究生，博士后研究人员和研究访问者积极参与该项目的工作。特别是，来自数学科学中代表性不足的群体的本科生受到指导并参与该项目不可或缺的研究经验，这些问题在数值和分析上都适合于学习和适合学习和部分进步的问题。该项目是在三个块中结构的，第一个块涉及不可压缩的欧拉方程以及其他水浪模型。在表面准地藻（SQG），广义SQG方程和相关模型上的第二个；该奖项反映了NSF的法定任务，这是对光谱几何学和数学物理学的应用的第三个，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估的。