Computing with Rational Functions
Rational functions are a mainstay of computational mathematics. As a result of recent breakthroughs, however, rational functions are now poised to become a central computational mathematics tool.
With this CAREER award, Alex Townsend, Mathematics, is pursuing two distinct aims related to the use of rational functions in computation. The first part of this project will develop data-driven algorithms for signal processing that include tools for filtering, feature detection, and super-resolution. This research has medical applications and could result in new feature-detection tools that are useful for classification tasks, such as detecting arrhythmia and seizures from electrocardiogram (ECG) signals. Processing ECG signals with rational-based super-resolution could prove to be as revolutionary as compressed sensing was for magnetic resonance imaging (MRI).
The second part of the project 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 toward gaining mechanistic understanding from experimental data in active fluids and advection-dominated flows. By providing a basis for rational neural networks that can discover differential equations with unprecedented accuracy, this research could lead to new models of complex fluids. Such models, developed from experimental data, will contribute to the understanding of active fluids and could potentially support the creation of biological turbines.