CAREER: Computing with Rational Functions

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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.