Current PhD students

A student working with a supervisor on a whiteboard

Studying with MARS

The MARS PhD programmes in Applied Mathematics and Mathematical AI empower our students to drive advancements at the interface of dynamical mathematical modelling and AI methods.

With a focus on the mathematical innovation required to address long-standing hard problems, our PhD students will make meaningful contributions to a rapidly evolving field of research.

Through close collaboration with industry, government, and academic partners, their work will have the potential to deliver significant real-world impact.

Current students

Gabriel Diaz-Aylwin

ML-Driven Design for Fusion Reactors

Gabriel Diaz-Aylwin

Lead supervisor: Dr Henry Moss

I work on data-aware engineering design optimisation: the development of techniques which, given a well-posed objective function and set of constraints, automatically search for improved designs.

Two paradigms are well established. In PDE-constrained optimisation, one solves the governing equations of the physical system and derives adjoint equations to compute sensitivities of the objective function. In practice, however, the governing equations may be approximate and incorporating model uncertainty into the optimisation loop is difficult. Furthermore, purely local exploration of infinite-dimensional design spaces typically leads to poor local optima.

At the other extreme, Bayesian optimisation treats the objective function as a black box, encoding limited assumptions through a kernel. While this approach is flexible and data-efficient, it makes only crude use of the often substantial physical and geometric knowledge available about the system.

This situation can be characterised as a tension between exploitation and trust: PDE-based methods exploit structure but demand confidence in the model, while Bayesian optimisation reduces trust requirements but discards structure. My research explores this trade-off. My interests span uncertainty quantification, discrete differential geometry, surrogate modelling of physical processes, and automatic differentiation.

My project is part-sponsored by the UK Atomic Energy Authority. The motivating application being design optimisation in the challenging physical environment of tokamak fusion reactors. In particular, we aim to address a key unknown for the MAST Upgrade divertor: how its performance depends on the equilibrium magnetic configuration and on fuel and impurity injection locations. By characterising this dependence, we aim to inform divertor design and operation.

Arthur Fearn-Rice

Agent Based Modelling of Wound Healing

Arthur Fearn-Rice

Lead supervisor: Dr Alice Peng

Wound healing is a complex biological process that can be studied and described mathematically using a variety of modelling techniques. This project focuses on agent-based modelling, a computational approach in which individual entities, cells, are represented as agents with defined properties such as location, shape, and size. Each cell interacts with both its environment, including the extracellular matrix (ECM), and with other cells within the modelled tissue. By simulating these interactions, we aim to gain a deeper understanding of wound healing dynamics, particularly the migration of cells both individually and collectively.

This approach allows us to apply tools from mathematics, statistics, and artificial intelligence to investigate key biological questions. For example, we explore:

  1. The dynamics of cell-cell adhesion and repulsion, and how these can be modelled effectively.
  2. Modelling distinct layers of skin, such as the dermis and epidermis, and defining realistic boundary conditions.
  3. Differentiating between healthy and infected tissue regions to better capture cell migration into wound sites.
  4. The computational trade-offs between modelling each cell as a simple circular agent versus using a more detailed vertex-based representation, which could improve model accuracy but increase complexity.

By extending current models, my project seeks to provide quantitative insights into the mechanics of wound healing, informing both theoretical understanding and potential clinical applications.

Amos Keshet

Tracking Disease Spread in Real Time

Amos Keshet

Lead supervisor: Professor Chris Jewell

This project focuses on combining next-generation mathematics and AI to create fast, reliable, and accessible tools for monitoring and responding to real-world disease outbreaks more effectively. It addresses one of public health’s biggest challenge - transforming incomplete, delayed, and complex data into accurate and actionable insights that can support rapid outbreak response.

Traditional disease monitoring systems often struggle to keep up with fast-moving outbreaks such as influenza, COVID-19, and measles, as reported cases frequently lag behind actual infections. This project leverages stochastic dynamical models and Monte Carlo methods (including Markov chain and Sequential Monte Carlo algorithms) to overcome these limitations. These models capture the uncertainty and variability in disease transmission and allow continuous calibration as new data becomes available.

By integrating data on location, demographics, socioeconomic factors, and hygiene practices, the models can estimate infection risks and identify population groups most vulnerable to rapid spread or antimicrobial resistance.

The research also focuses on building user-friendly tools that make these complex models more accessible to public health teams, enabling faster and more effective decision-making during outbreaks.

Ultimately, my project aims to modernize outbreak modelling, using next-generation mathematics and AI to provide a dynamic, evidence-based framework for responding to infectious diseases.

The overall goal is to use these methods to produce practical software tools that:

  • Simplify modelling.
  • Enable rapid model development.
  • Lower the barrier for researchers and public health professionals to use outbreak modelling.
Dawid Lipinski

Optimal Transport Methods in Machine Learning

Dawid Lipinski

Lead supervisor: Dr Maciej Buze

Minimisation diagrams and optimal transport provide powerful frameworks for modelling the microstructure of materials such as steel and glass. Achieving high accuracy in these models often requires the optimisation of millions of parameters, making traditional numerical methods computationally expensive. Recent advances in machine learning, particularly GPU-optimised methods implemented through modern Python libraries, have demonstrated substantial speed-ups for these optimisation tasks.

Building on this progress, my PhD project aims to extend existing approaches to the more general setting of unbalanced semi-discrete optimal transport. Unbalanced optimal transport allows for the comparison of distributions with differing total mass, which is essential when modelling a wide range of physical phenomena. This extension is expected to enable more accurate and flexible modelling of material microstructures while preserving the computational efficiency offered by machine-learning-based methods.

In parallel, the project explores connections with diffusion models, which have recently achieved remarkable success in areas such as image and video generation, healthcare, and biology. Diffusion models can be interpreted as describing the evolution of measures or distributions, closely relating them to unbalanced optimal transport. By studying this connection in greater depth, my research seeks to broaden the theoretical understanding of diffusion models and expand their range of practical applications.

Abiel Talwar

Guiding Scientific Search with Generative AI: Bayesian Optimization with Diffusion Priors

Abiel Talwar

Lead supervisor: Dr Henry Moss

Machine learning (ML) guided design has sparked transformative change in experimental methodologies across various scientific fields, from discovering novel biological sequences to engineering innovative engineering structures and even developing superior ML algorithms. Generative AI methods, particularly diffusion models and flow matching, have recently emerged as powerful tools for creating complex, highly structured outputs like images, molecules, and point clouds. Consequently, these models hold tremendous potential for accelerating resource-intensive design processes.

Despite their promise, integrating generative AI into ML-guided design loops, such as Bayesian optimization and active learning, is a significant challenge due to the inherent difficulty in updating pre-trained generative models to incoming new data. In these settings, the goal is to fine-tune the generation process iteratively, producing designs that increasingly align with target objectives and design constraints. I shall be exploring new techniques for fine-tuning conditional sampling, enabling diffusion models to be more effectively integrated into optimization pipelines.

This PhD is funded by a philanthropic donation from GResearch.

Inayat Ullah

Numerical Continuation and Deflation Techniques for Atomistic Modelling of Materials

Inayat Ullah

Lead supervisor: Dr Maciej Buze

This research addresses a fundamental computational challenge of precisely locating equilibrium points in atomistic energy landscapes, which govern critical material phenomena such as fracture and the nucleation and propagation of crystal defects. Mathematically, this challenge can be framed as a root-finding problem, where we seek atomic configurations that satisfy the force balance equation, ensuring that the net force is zero.

Our methodology employs advanced numerical analysis techniques, leveraging bifurcation theory and numerical continuation to systematically trace entire curves of equilibrium solutions, including both stable minima and mechanically unstable saddle points that are critical for reaction pathways. My initial foundational work focuses on optimizing Newton's method to efficiently identify stable configurations. Additionally, the project will integrate deflation techniques to systematically uncover multiple distinct solution branches.

The long-term objective is to implement a Hessian-free continuation approach to overcome the significant computational scaling limitations associated with traditional methods. The resultant computational framework aims to serve as a powerful new tool, validated against problems such as mode-I and mode-III fracture and dislocation nucleation. Our goal is to establish this integrated method as a standard, versatile technique for materials science research, providing a robust framework for advanced materials modelling and generating improved predictivemodels for material failure.

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