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CAREER: Computing with Rational Functions

职业：使用有理函数进行计算

基本信息

批准号：
2045646
负责人：
Alex Townsend
金额：
$ 50万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2045646&HistoricalAwards=false
关键词：
CAREER Computing Rational Functions

项目摘要

Rational functions are a mainstay of computational mathematics, and due to several recent breakthroughs, they are now ready to go from a specialized field to a central computational mathematics tool. The project has two main aims: (1) Providing new feature detection tools in signal processing that are useful in classification tasks, including arrhythmia and seizure detection from ECG signals and (2) Using rational neural networks to get unprecedentedly accurate partial differential equation discovery, leading to new models of complex fluids. It is possible that rational-based superresolution in signal processing for ECG signals is an idea that could be as revolutionary as compressed sensing for MRI imaging. Developing new models from experimental data will contribute to the heated debate on how active fluids are modeled, potentially leading to biological turbines. An educational component of the project includes a novel undergraduate curriculum design; outreach to the local high schools; writing a textbook, and creating online YouTube lecture courses. Training of graduate students will be integrated in the project.In the first part of this project, the PI will develop data-driven algorithms for signal processing, including tools for filtering, feature detection, and superresolution. The PI will tackle open problems related to understanding the convergence behavior of these adaptive algorithms with consequences in model reduction and nonlinear eigensolvers. In the second part of the project, the PI will use rational neural networks in deep learning to develop an approach that rigorously discovers Green's function associated with elliptic partial differential equations from data. This will be a step towards gaining mechanistic understanding from experimental data in active fluids and advection-dominated flows.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.

有理函数是计算数学的支柱，由于最近的一些突破，它们现在已经准备好从一个专业领域转变为一个核心计算数学工具。该项目有两个主要目标：（1）在信号处理中提供新的特征检测工具，可用于分类任务，包括从心电图信号中检测心律失常和癫痫发作；（2）使用理性神经网络获得前所未有的精确偏微分方程发现，从而产生复杂流体的新模型。 ECG 信号信号处理中基于理性的超分辨率可能与 MRI 成像的压缩传感一样具有革命性。根据实验数据开发新模型将有助于引发关于如何模拟活性流体的激烈争论，并有可能催生生物涡轮机。该项目的教育部分包括新颖的本科课程设计；向当地高中推广；编写教科书，创建在线 YouTube 讲座课程。研究生的培训将纳入该项目。在该项目的第一部分，PI 将开发数据驱动的信号处理算法，包括滤波、特征检测和超分辨率工具。 PI 将解决与理解这些自适应算法的收敛行为相关的开放问题，以及模型简化和非线性特征求解器的后果。在该项目的第二部分中，PI 将在深度学习中使用理性神经网络来开发一种方法，从数据中严格发现与椭圆偏微分方程相关的格林函数。这将是从活跃流体和平流主导流的实验数据中获得机械理解的一步。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。