Research

I am interested in using the tools of mathematics (especially dynamical systems and probability theory)  to apply machine learning to problems in science and engineering. In my work I like to focus on the follow questions:

• Why does machine learning work? We cannot hope to apply machine learning to problems in which errors lead to significant consequences (e.g medicine, some engineering applications) without an understanding of when and why machine learning algorithms fail.  Mathematical insight into machine learning will help to address these questions.

• Many top performing machine learning approaches do not produce interpretable models, which is an overarching goal of science. How can we use machine learning  to speed up the process of scientific discovery?

Current Work

I am finishing up a project in which we study the behavior of a randomly perturbed intermittently chaotic dynamical system through properties of the associated transfer operator. To access information about the transfer operator,  we have adapted Extended Dynamic Mode Decomposition (EDMD) to compute a matrix representation of the transfer operator. Below are plots of some invariant densities (left) and spectra (right) at different noise levels.