School of Mathematics & Statistics

Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.

Structured ML for physical systems (PhD or MSc)

Supervisors: Lawrence Bull
Relevant research groups: Machine Learning and AI, Emulation and Uncertainty Quantification

When using Machine Learning (ML) for science and engineering, an alternative mindset is required to build sensible representations from data. Unlike other applications (e.g. large language models), the datasets are relatively small and curated - i.e. they are collected via experiments rather than scraped from the internet. The limited variance of training data typically renders learning by ‘brute force’ infeasible. Instead, we must encode domain-specific knowledge within ML algorithms to enforce structure and constrain the space of possible models.

This project covers ML for physical systems (Karniadakis, 2021) and looks to integrate machine learning with applied mathematics - fusing scientific knowledge with insights from data. Methods will investigate various levels of constraints on ML predictions - including smoothness, invariances, etc. Relevant topics include:

• physics-informed machine learning (Girin, 2020)
• scientific machine learning (Pförtner, 2022)
• hybrid modelling

Generating deep fake left ventricles: a step towards personalised heart treatments (PhD)

Supervisors: Andrew Elliott, Vinny Davies, Hao Gao
Relevant research groups: Machine Learning and AI, Emulation and Uncertainty Quantification, Biostatistics, Epidemiology and Health Applications, Imaging, Image Processing and Image Analysis

Personalised medicine is an exciting avenue in the field of cardiac healthcare where an understanding of patient-specific mechanisms can lead to improved treatments (Gao et al., 2017). The use of mathematical models to link the underlying properties of the heart with cardiac imaging offers the possibility of obtaining important parameters of heart function non-invasively (Gao et al., 2015). Unfortunately, current estimation methods rely on complex mathematical forward simulations, resulting in a solution taking hours, a time frame not suitable for real-time treatment decisions. To increase the applicability of these methods, statistical emulation methods have been proposed as an efficient way of estimating the parameters (Davies et al., 2019; Noè et al., 2019). In this approach, simulations of the mathematical model are run in advance and then machine learning based methods are used to estimate the relationship between the cardiac imaging and the parameters of interest. These methods are, however, limited by our ability to understand the how cardiac geometry varies across patients which is in term limited by the amount of data available (Romaszko et al., 2019). In this project we will look at AI based methods for generating fake cardiac geometries which can be used to increase the amount of data (Qiao et al., 2023). We will explore different types of AI generation, including Generative Adversarial Networks or Variational Autoencoders, to understand how we can generate better 3D and 4D models of the fake left ventricles and create an improved emulation strategy that can make use of them.

Regular seminars relevant to the group are held as part of the Statistics seminar series. The seminars cover various aspects across the AI3 initiative and usually span multiple groups. You can find more information on the Statistics seminar series page, where you can also subscribe to the seminar series calendar.