Student
Dáire O’Kane

Dáire graduated from the University of Edinburgh in 2023 with an MMath Mathematics degree

Dáire graduated from the University of Edinburgh in 2023 with an MMath Mathematics degree. He mainly studied analysis, algebra and stochastic analysis. His masters dissertation was in Gaussian analysis, exploring the properties of Gaussian measures in infinite dimensions and their role in constructive quantum field theory.

Dáire is completing a PhD project with a focus on stochastic differential equations, particularly those relevant in data science. This includes investigating the numerics of SDEs and their connections with deep learning.

In his spare time, Dáire enjoys yoga, climbing, reading, and trekking up a good hill.

Project title: 
Applications of Stochastic Differential Equations in Data Science

Supervisor(s):
James Foster, Chris Budd

Project description:
In recent years, the theory of stochastic differential equations (SDEs) has been directly connected to several breakthroughs in machine learning. This increasingly fruitful relationship between machine learning and SDEs has produced a variety of state-of-the-art techniques, ranging from diffusion models in computer vision to Langevin algorithms for Bayesian learning. At the same time, SDEs are themselves classical models from mathematics and commonly used to describe random phenomena such as molecular dynamics, biological systems and financial markets. Consequently, there is significant interest in applying the computational tools from machine learning to enhance traditional SDE modelling.

The project sits at the intersection of SDE theory and data science, with the aim of generating innovative research across the two domains. This includes investigating numerical methods and associated convergence rates for solving Langevin dynamics, an SDE system popular in both molecular dynamics and machine learning communities, which cannot be solved analytically. We are also interested in whether recently developed SDE solvers can improve popular learning algorithms, such as Stochastic Gradient Langevin Dynamics – which is based on Euler’s method.