Research

Machine learning and dynamics

In this project, developed with Prof Kühn, we explore the behaviour of machine learning, starting from RNNs, from the point of view of dynamical systems using tools from validated numerics. The first results can be found in this article and this code

We are also working on relationships between chaos and RNNs through the study of Lyapunov exponents.

Network dynamics in cell biology

My collaboration with Dr Kepley and Prof Mischaikow has brought me to work on dynamics defined on networks, such as the Toggle-Switch. We have a paper tracking and understanding saddle nodes in high-dimensional parameter spaces

In this topic, I was co-organizer of the .

Bifurcations

Tracking where the dynamics of a system changes qualitatively. In collaboration with Prof van den Berg and Prof Lessard, saddle nodes and Hopf bifurcations can now be validated in polynomial ODEs. You can find the article and the code. An application to PDEs can be found in this article.

In an article with Dr Church, we explored codimension 2 bifurcations, 2D manifolds of solutions and delay differential equations. The BiValVe library is associated.

Validated numerics

Turning a numerical result into a rigorous proof is the key element of this section. My PhD was focused on validated numerics, in particular following branches of periodic orbits, includes a library. Work with Prof van den Berg.

Eigenvalues

Related to bifurcations, is the topics of eigenvalues: how do they change with respect to a parameter? How can we validate them?

I have been collaborating with Prof Plum on determining the eigenvalues of a slid disc, depending on the length of the slid.