About me and my research Link to heading

I am Davide Murari, and I am from Italy. I like reading, long-distance running and trying to explain what I know online. I have a Youtube channel about mathematics’ popularisation in Italian, that can be found here Youtube Channel I’m a Postdoctoral Research Associate in the Cambridge Image Analysis (CIA) group at the University of Cambridge.

My PhD thesis can be found here and is titled “Neural Networks, Differential Equations, and Structure Preservation”. I have completed my PhD in September 2024, in the Differential Equations and Numerical Analysis group at NTNU, Trondheim, Norway, under the supervision of Elena Celledoni and Brynjulf Owren.

Both my Bachelor degree and my Master degree were in Applied Mathematics, at the University of Verona, Italy. My bachelor thesis was on dynamical billiards, while the master’s one on the theory of integrability of non-Hamiltonian dynamical systems.

During university, I developed a great interest in dynamical systems and geometric mechanics. In the DNA group at NTNU, I found the opportunity of merging these two interests with the one of numerical implementation. My PhD thesis will focus on

analysing deep neural networks from the perspective of dynamical systems and data-driven modelling for dynamical systems.

By dynamical systems’ approach to deep learning, I refer to their possible interpretation as non-autonomous parametric ODEs. Indeed, this comes thinking to neural networks having infinitely many layers, where time is considered a measure of the depth of the network, having hence infinitely many layers.  Therefore, for example, we can think of the challenge of binary classification of points of the plane as “learning a vector field whose flow moves the points so that a hyperplane can separate the two labelled groups”.

Thanks to this construction, many relevant questions and techniques typical of ODEs and Numerical analysis arise in this research area and make me interested in these problems.

Such a powerful connections between these fields, goes also in the opposite direction. Indeed, with the increasing amount of data we are able to collect nowadays, it becomes interesting to answer to the two following questions - can we approximate the vector fields generating a set of measured trajectories? - can we approximate the solutions to PDEs and ODEs using hybrid approaches involving data driven techniques and numerical analysis? During my PhD I am to investigate these two questions from different angles, and you can already see some work in this direction in the Academic Works page of the website.

I created this personal website to keep track of my research work, improve my writing and share what I learn during the process in the page Notes . I hope to get in touch with many other researchers interested in the topics on which I am working, possibly starting some nice collaborations.