CAREER: Computing with Rational Functions

有理函数是计算数学的支柱，由于最近的几项突破，它们现在已经准备好从一个专业领域发展成为一个中心的计算数学工具。 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，用于分类任务，包括从ECG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。在ECG信号的信号处理中，基于理性的超分辨率可能是一个与MRI成像的压缩传感一样具有革命性的想法。根据实验数据开发新模型将有助于对如何模拟活性流体进行激烈的辩论，可能导致生物涡轮机。该项目的教育部分包括一个新颖的本科课程设计;推广到当地高中;编写教科书，并创建在线YouTube讲座课程。在该项目的第一部分，PI将开发用于信号处理的数据驱动算法，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，这些算法在模型简化和非线性特征解算器中具有后果。在项目的第二部分，PI将在深度学习中使用理性神经网络来开发一种方法，该方法可以从数据中严格发现与椭圆偏微分方程相关的绿色函数。这将是从活跃流体和对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。

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.