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，在寻找具有有限能量的不可压缩流体中的奇点或整体存在的最新工作中发展的方法和思想，另一方面，将技术转移到解决光谱几何和数学物理中长期存在的开放性问题。其成功的一个关键因素是精确的数值，以及对规律性理论的深刻理解。因此，该项目的跨学科性质，包括数值计算，物理通知神经网络（PINN），计算机辅助证明，偏微分方程（PDE）的现代方法和谐波分析，是其成功结果的重要组成部分。最近出现的将数学分析和深度学习相结合的技术为开发新的攻击角度和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、研究访问学者积极参与本项目的工作。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到项目的研究经验中，这些研究经验涉及丰富且适合学习和部分进步的问题，包括数值和分析。该项目分为三个部分，第一部分涉及不可压缩欧拉方程以及其他水波模型；二是关于地表准地转（SQG），广义SQG方程，及相关模型；第三个是关于光谱几何和数学物理的应用。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。