Lead supervisor: Dr Catherine Drysdale (c.drysdale@lancaster.ac.uk)

Depression is a heterogeneous disorder, with subtypes such as melancholic and atypical depression exhibiting distinct hormonal and neural signatures. The hypothalamic-pituitary-adrenal (HPA) axis plays a pivotal role, with melancholic depression often marked by elevated cortisol and atypical depression by hypocortisolemia. Understanding these dynamics requires tools capable of quantifying system sensitivity and instability beyond what standard eigenvalue analysis can capture.

This project will aim to create models of a wider brain network and hormonal system to model depression’s underlying mechanisms and treatment responses. The resulting time series of the models will be analysed via the technique of pseudospectra. Pseudospectra extend spectral analysis by revealing transient amplification and sensitivity to small perturbations, which are highly relevant for hormonal oscillations and brain network dynamics. By embedding structured perturbations into models of the HPA axis and corticolimbic circuitry (hippocampus, amygdala, and prefrontal cortex), the research will simulate stress responses and therapeutic interventions

Lead supervisor: Dr Catherine Drysdale (c.drysdale@lancaster.ac.uk)

Electroencephalography (EEG) provides a non-invasive, cost-effective means of capturing brain activity with high temporal resolution. However, EEG signals are notoriously noisy, influenced by artifacts from muscle activity, eye movements, and environmental interference, as well as the intrinsic complexity of neural dynamics. Extracting mechanistic insights from such data requires sophisticated mathematical tools that can disentangle meaningful structure from noise.

Dynamic Mode Decomposition (DMD) is a data-driven method that decomposes time-series into spatiotemporal modes, linking data to underlying dynamical systems. DMD has been successful in fluid dynamics and neuroscience applications such as fMRI, yet its application to EEG remains underdeveloped due to two major challenges: (i) the low signal-to-noise ratio, which destabilizes mode estimation, and (ii) the difficulty of linking transient, multi-frequency oscillations to clinically meaningful states.

This project proposes to extend and innovate DMD methodology to address these challenges. By integrating robust noise modelling, regularization strategies, and multi-scale decomposition, the research will create algorithms capable of extracting stable and physiologically meaningful modes from EEG. These innovations will be validated against benchmark datasets and aligned with known neural rhythms (theta, alpha, beta, gamma) to assess their clinical and cognitive relevance.

Ultimately, this work aims to establish a principled framework for using DMD on noisy EEG data, bridging the gap between mathematical modelling and practical neuroscience. Beyond methodological advances, the project has the potential to uncover new biomarkers of brain function, enabling more precise monitoring of mental health, neurological disease, and cognitive performance.

Lead supervisor: Dr Murad Banaji (m.banaji@lancaster.ac.uk)

A huge challenge in modern mathematical biology and medicine is to understand how complex biological networks robustly perform certain functions, and how this functioning is disrupted in disease. As network size increases, mapping out the allowed dynamics of a network across parameter space using computational algebra and brute-force numerical simulation alone rapidly becomes unfeasible.

A complementary approach is to develop results which tell us how large networks "inherit" behaviours from their smaller subnetworks which are easier to analyse more completely. This approach, based on dynamical systems theory, including bifurcation theory, has proved very fruitful in recent years. It has also shed light on "emergent" properties of biological networks: the process by which interacting subnetworks can give rise to new dynamical behaviours not seen in any individual subnetwork.

The aims of this PhD project are twofold:

(i) To develop further the theory of inheritance and emergence in biological networks, and apply it to make claims about signalling/metabolic networks of biological importance. This work would shed further light on the relationship between network structure and network function, and the mechanisms of disease.

(ii) To develop computational tools, including machine learning tools, to aid in the analysis of large biological networks via subnetwork decomposition. These would include tools to give descriptions of allowed dynamical behaviours, the parameter regions where they occur, their genesis from subnetworks, and their sensitivity to network modifications. This work would take an important step towards automating the dynamical analysis of biological networks of real-world scale.

Lead supervisor: Professor Andrew Baggaley (a.baggaley1@lancaster.ac.uk)

Agent-based models (ABMs) are a powerful tool for describing complex systems in which individual agents interact to produce emergent population-level behaviour. Traditional ABMs, however, are often difficult to calibrate against experimental data due to their non-differentiable structure. This project will develop differentiable ABMs to model the growth and treatment response of cancer tumours, enabling the use of gradient-based optimisation and machine learning techniques to directly infer mechanistic parameters from experimental data.

Working with experimental collaborators, models will be calibrated using comprehensive data from the growth of 3D cancer spheroids, a physiologically relevant in vitro system for studying tumour growth and therapeutic response. Examining both spheroid growth dynamics and the response of tumours to treatment will provide an unprecedented opportunity to develop mathematically rigorous models that capture processes such as cell proliferation and motion, apoptosis, spatial competition, and drug-induced perturbations.

Differentiable ABMs are a powerful paradigm that allows efficient parameter inference and the quantification of model uncertainty. A key focus of the project will be the development of efficient simulation algorithms that remain differentiable while faithfully representing spatially explicit, stochastic cellular dynamics. This work has the potential to provide new insight into tumour growth mechanisms, while also advancing the mathematical foundations of differentiable simulation. It will particularly suit candidates interested in applying applied mathematics, computation, and data-driven modelling to cutting-edge problems in biological research.

Lead supervisor: Professor Andrew Baggaley (a.baggaley1@lancaster.ac.uk)

The interiors of neutron stars contain matter at extreme densities, where neutrons are believed to form a superfluid, an exotic state of matter where vorticity is constrained to space-curves called quantised vortices. The motion and interaction of these vortices is central to explaining astrophysical phenomena such as pulsar glitches, spasmodic jumps in the observed rotation rate of the star. In particular, a key problem that motivates this project is to understand the interaction of vortices in the presence of a landscape of pinning sites, which we believe is the case in the star’s inner crust. Directly probing these dynamics is not possible at present, so laboratory analogues play a vital role. Recent experiments in superfluid helium-3, where an aerogel is embedded to create a complex pinning landscape, provide a unique platform for studying vortex dynamics, and the pinning and unpinning processes analogous to those expected in neutron stars.

This PhD project will develop mathematical and computational models of these systems using the Gross–Pitaevskii equation (GPE), a nonlinear Schrödinger equation widely used to describe superfluid dynamics. The candidate will build high-resolution simulations of vortex motion, and pinning in complex landscapes, with the dual aims of (i) reproducing and interpreting laboratory experiments in helium-3–aerogel systems, and (ii) transferring insights to the astrophysical context of neutron stars.

The project will combine rigorous PDE modelling, numerical analysis, and data-driven approaches. Differentiable and machine learning–based inference tools will be explored to fit GPE-based simulations to experimental data, bridging the gap between microscopic vortex dynamics and macroscopic observables such as glitch statistics.

Lead supervisor: Dr Alice Peng (qiyao.peng@lancaster.ac.uk)

Traction force microscopy (TFM) is a technique for measuring the traction forces generated by a moving cell on a substrate or surface [1]. In its more basic setting, the cells are placed on an elastic substrate, that contains fluorescent beads. When the cell moves, it applies a force to the substrate, producing a displacement that can be measured by observing the positions of the fluorescent beads. TFM has many applications, for example, traction forces can be used as a proxy to estimate the invasive potential of a cancer cell line.

The aim of this project is to extend the current data analysis methodologies from TFM data with the use of better mechanical models; vice versa, the data is expected to improve the model to describe the mechanical cell-ECM interactions. As the project focuses on single cells, agent-based modelling is favoured. Peng et.al. [2] proposed a phenomenological model, in which the cell geometry is split into finite line segments by nodal points. By doing this, the model benefits from the flexibility of modelling \textit{any} geometry, which is relevant to the forces generated by the cell. Instead of a smooth force field, point forces are used in this setting, which allows more accurate estimates of the force. In this project, the successful candidate will use physics-informed neural networks and scientific machine learning tools (e.g. SINDY) to develop novel parameter estimation methodologies for the models and to improve the computational efficiency of solving the system.

A suitable candidate for the project will have a strong background in applied mathematics and coding skills, and an interest in the areas of inverse problems or statistical inference.

References
[1] A. K. Denisin, H. Kim, I. H. Riedel-Kruse, and B. L. Pruitt. Field guide to traction force microscopy. Cellular and Molecular Bioengineering,17(2): 87–106, Apr. 2024. ISSN 1865-5033. doi: 10.1007/s12195-024-00801-6.
[2] Q. Peng, F. J. Vermolen, and D. Weihs. A formalism for modelling traction forces and cell shape evolution during cell migration in various
biomedical processes. Biomechanics and Modeling in Mechanobiology, Apr. 2021. ISSN 1617-7940. doi: 10.1007/s10237-021-01456-2

Lead supervisor: Dr Alice Peng (qiyao.peng@lancaster.ac.uk)

Cells proliferate by passing through the cell-cycle, where they undergo phases of growth, replicate their DNA and finally split into two daughter cells. Cancer cells overcome the normal tight regulation of this process to proliferate in an uncontrolled manner. This uncontrolled growth and the metastasis of cancer cells are two of the key hallmarks of cancer leading to poor patient outcomes.

The data used to build and test cell-cycle models is often from techniques that analyse bulk populations of cells. For this reason, the effects on tumour growth of inter-cell heterogeneity, and the link between the cell-cycle and migration, remain understudied. Both of these features have key bearings on the effectiveness of different therapeutics, and a deeper understanding of these aspects of the cancer cell-cycle is crucial to improving treatments.

For this project, we have access to unique datasets tracking 1000s of individual cancer cells through the different stages of the cell-cycle, whilst simultaneously tracking their position. This data gives unprecedented insights into cell-cycle heterogeneity and the relationship between the cell-cycle and cell migration.

The candidate will use a variety of modelling methodologies to develop mathematical models of cancer cells. These will include both deterministic and stochastic models, and will consider not only the phases of the cell-cycle but also the effects of cell heterogeneity and cell-cell interactions. The available experimental data will be used alongside machine learning methods to aid in model parameterisation and validation. Model dynamics will be explored to gain insight into the links between cell-cycle heterogeneity, cell proliferation and cell migration.

Lead supervisor: Dr Jess Bridgen (j.bridgen@lancaster.ac.uk)

Calibrating infectious disease models to real-time data remains a key computational and methodological challenge in outbreak response, representing a gap between the possibilities presented by such models, and their use in the real-world to inform policy. Bridging this gap is crucial to providing accurate and timely predictions of outbreak trajectories, evaluating control measures, and testing of control strategies.

In the project, the student will develop stochastic dynamical models and inference methodology, utilising high performance computing techniques to calibrate these high-dimensional models in real time. Alongside model development, the candidate will create a modular software library for infection control. This software will enable users to design their own control strategy from a set of pre-defined options and assess their effectiveness. Strategies may include testing protocols, isolation measures and vaccination programmes.

The models and software will be applied in hospital settings, working with NHS partners to develop data-driven approaches for infection control. Hospital outbreaks are a major challenge for infection control and require targeted control measures to be implemented rapidly. Current models in these settings are often constrained by computational complexity and scalability as they need to explicitly represent the complex hospital environment, including ward layouts and the network of staff-patient interactions.

Objectives:
(1) Construct individual-level stochastic dynamical models of transmission for different respiratory pathogens.
(2) Develop methodology to calibrate stochastic transmission models in real-time.
(3) Create modular software to simulate and evaluate infection control strategies.

This project would suit candidates with an interest in dynamical modelling, high performance computing, and solving public health challenges.

Lead supervisor: Dr Mher Safaryan (mars@lancaster.ac.uk)

Mathematical optimization is one of the main engines behind modern machine learning. Its goal is to adjust a model’s parameters so that it performs well on training data, typically by minimizing prediction errors. As models (such as deep neural networks) and datasets have grown dramatically in size, the computational and energy costs of training and deploying them have also exploded. This creates new challenges and opportunities for optimization research.

One direction focuses on model compression, which reduces the size of large models while preserving accuracy. Because large models often contain redundancies, they can frequently be compressed with little or no loss in performance. Two key techniques are sparsification (setting some parameters to zero) and quantization (using fewer bits to represent each parameter). The main challenge is to design optimization algorithms that remain effective under such compression constraints, whether applied during training (compression-aware training) or after training (post-training compression). This also motivates revisiting the design of optimization algorithms themselves. While Adam (a variant of stochastic gradient descent) remains the default choice in deep learning, newer optimizers such as Shampoo, SOAP, and Muon show promising alternatives. This direction aims to make training and inference more energy-efficient, enabling AI models to run not only on large cloud-based servers but also on smaller, resource-constrained devices such as smartphones.

A second direction addresses the challenge of scaling data through federated learning. Here, many clients (e.g., mobile devices, hospitals, IoT systems) collaborate to train a shared model without directly sharing their local data, thus preserving privacy. Compared to centralized training, this introduces new optimization challenges: how to reduce communication between clients, how to handle asynchronous updates efficiently, and how to ensure robustness against adversarial participants.

Together, these two directions explore how to make machine learning optimization more scalable, efficient, and trustworthy, developing both theoretical convergence guarantees and practical methods that enable powerful models to be trained and deployed with lower cost and higher reliability.

Lead supervisor: Dr Jixiang Qing (j.qing@lancaster.ac.uk)

Sequential decision making has been commonly used in Bayesian optimisation, reinforcement learning and control. Decisions are made given the current state of the system and can affect the future behaviour of the system, hence they need to be wisely chosen to maximize a pre-specified reward.

However, in Bayesian optimisation, traditional policies are typically hand-crafted, while learning-based approaches can demonstrate stronger empirical performance but either need huge amounts of interaction with environments or can only be deployed in very similar environments to achieve satisfactory performance, making them risky to use in many situations. Besides, most sequential decision-making approaches cannot handle complex trajectory constraints (e.g., planning optimal fuel-efficient routes through unknown terrain while maintaining safety reserves, or optimizing chemical reaction pathways while avoiding catalyst-poisoning conditions that would terminate the process), thus limiting their applicability in safety-critical fields.

This PhD research project will investigate novel algorithms that provide safety guarantees for sequential decisions. This includes developing performance-guaranteed policies that can be deployed in new environments while handling complex real-life constraints (e.g., catalyst states, resource budgets), bridging the gap between the empirical success of learning-based policies and the reliability of traditional approaches.

Prospective students are expected to become familiar with: Markov decision processes, Bayesian optimisation, reinforcement learning, and (optionally) generative AI as powerful policy models.

Lead supervisor: Dr Jixiang Qing (j.qing@lancaster.ac.uk)

Differential equations provide the mathematical foundation for modelling temporal phenomena across diverse domains, from physical processes like chemical reactions to modern generative AI. Recent progress in data-driven dynamical systems models (e.g., Neural ODE/SDE, Gaussian Processes-based vector field modelling, State-space models) has created new possibilities for sequential decision making in such systems, which is of practical importance in various domains. For instance, in chemical reactions, one is interested in identifying the optimal reactant, as well as additional continuous signals (temperature, the flow rate of reactants) to achieve maximum product yield at the end of the reaction process.

However, several gaps still exist that prevent fully realising this potential: efficient and flexible modelling of unknown dynamical systems from limited multivariate timeseries data remains an open issue; scalable optimisation methods for high-dimensional control signals are computationally challenging; and most model-based approaches require good initial models, creating a problematic two-stage separation between model learning and policy learning.

This PhD proposal focuses on two interconnected aspects: developing efficient probabilistic inference methods for unknown dynamical systems with limited data, and designing sequential decision-making algorithms (control signal) that optimise performance. The key research challenge is investigating approaches to unify these traditionally separate phases (e.g., through online probabilistic inference methods) that can adapt in real-time as decisions are made.

Prospective students are expected to learn to become familiar with dynamical system models, probabilistic inference (e.g., Sequential Monte Carlo), and model predictive control/model-based RL/dual control.

Lead supervisor: Dr Matthias Sachs (m.sachs@lancaster.ac.uk)

Coarse-graining (CG) is a dimensionality reduction technique used to reduce the complexity of atomistic simulations by lowering the number of explicitly simulated degrees of freedom, for example, by grouping atoms into larger interaction sites thereby enabling simulations of much larger systems and longer timescales than all-atom models, making CG an essential tool for studying processes and properties in materials science and biochemistry that only emerge at these scales. The accuracy of predictions obtained from such simulations heavily depends on the form and the underlying construction of the coarse-grained model.

Traditionally, coarse-grained models are handcrafted based on physical or chemical intuition. While for simple systems, such coarse-grained models often perform well, their construction can become tedious, and generalisation poor as the complexity of the coarse-grained atomistic system increases. This project aims to develop end-to-end, data-driven machine learning approaches that enable automatic and systematic learning of coarse-grained models, along with numerical integration methods that facilitate computationally efficient simulation of these coarse-grained systems. A particular focus will be on approaches that faithfully preserve the dynamics of the original system. We aim to achieve this by combining well-founded mathematical theory on dynamics preserving coarse-graining, including the Mori-Zwanzig Formalism [1], with modern machine learning approaches, such as equivariant graph neural networks [2], which allow for flexible transformations of the degrees of freedom of the atomistic system, and elements of generative modelling [3].

During your PhD, you will extend your expertise in areas of mathematics relevant to this project, e.g., stochastic modelling, numerical analysis, and theory related to the Mori-Zwanzig Formalism. You will develop your scientific expertise in molecular modelling, and you will extend your expertise in theoretical and practical aspects of recent machine learning approaches. As part of this project, you will implement, systematically test, and evaluate the developed methods in Julia or Python. The code will be deployed as a software package to facilitate collaborations with groups in engineering and computational chemistry.

[1] Lin, Yen Ting, et al. "Data-Driven Learning for the Mori-Zwanzig Formalism: A
Generalization of the Koopman Learning Framework." (2021)
[2] Batatia, Ilyes, et al. "MACE: Higher order equivariant message passing neural networks for
fast and accurate force fields." (2022)
[3] Wang, Wujie, et al. "Generative coarse-graining of molecular conformations." (2022).

Lead supervisor: Dr Matthias Sachs (m.sachs@lancaster.ac.uk)

Generative modelling of 3D molecular structures has a wide range of applications in scientific domains where the development and discovery of new molecules that satisfy a prespecified set of properties are of interest. For example, in drug discovery, generative models are utilised to find new drug candidate molecules with prespecified binding affinity.

While common generative modelling approaches, such as generative diffusion models, flow matching, and self-normalising flows, have been shown to perform excellently in, for example, image generation tasks, the more complex nature of the 3-dimensional structure of molecules with inherent geometric symmetries, mixture of discrete features (e.g., chemical element types of atoms) and continuous features (e.g., atom positions) and various physical constraints pose additional challenges for the design of generative models for 3D molecular structures.

The primary goal of this project will be to develop new generative modelling approaches for 3D molecular structures based on equivariant graph neural networks. At the same time, there will also be opportunities for more foundational work where we study certain aspects of generative modelling approaches in a rigorous mathematical framework. This may include numerical aspects of (stochastic) differential equation-based generative models, such as generative diffusion models or score matching approaches, and the analysis of generative modelling approaches in an optimal transport framework.

Lead supervisor: Dr Maciej Buze (m.buze@lancaster.ac.uk)

Machine Learning Interatomic Potentials (MLIPs), such as Atomic Cluster Expansion (ACE) [1], provide a method to approximate the potential energy of systems of atoms a function of their positions. These approaches serve as computationally efficient alternatives to traditional quantum mechanical calculations, which also account for the complex behaviour of electron clouds surrounding atomic nuclei. By harnessing advanced machine learning techniques, MLIPs have the potential to achieve quantum-level accuracy at a significantly reduced computational cost. However, the current lack of robust uncertainty quantification (UQ) in these models limits their reliability, particularly in applications like materials discovery.

In this project, we aim to establish a robust UQ framework for MLIPs, utilising principles from approximation theory [2] and Bayesian inference [3]. The successful candidate will develop a comprehensive toolbox in either Julia or Python, applying it to various test scenarios to validate its effectiveness. Throughout this research, the student will gain extensive knowledge in modern numerical analysis, engage in coding a specialised library, and conduct GPU-accelerated molecular dynamics simulations. This position offers a unique opportunity to contribute to the ongoing AI-driven transformation in materials science, positioning the student at the forefront of innovative research in this exciting field.

[1] Drautz, R. (2019). Atomic cluster expansion for accurate and transferable interatomic potentials. Physical Review B, 99(1), 014104.
[2] Trefethen, L. N. (2019). Approximation theory and approximation practice, extended edition. Society for Industrial and AppliedMathematics.
[3] Barber, D. (2012). Bayesian reasoning and machine learning. Cambridge University Press.

Lead supervisor: Dr Henry Moss (henry.moss@lancaster.ac.uk)

Probabilistic machine learning methods have had a profound impact across the sciences, not only in data-driven scientific discovery but also in accelerating the development of core mathematical algorithms. This has given rise to the emerging field of probabilistic numerics, which views numerical computation itself as an inference problem. State-of-the-art solvers for differential equations, numerical integrators, and linear algebra increasingly rely on such ideas.

We are now entering the era of generative AI, where diffusion models, flow-matching methods, and related approaches are reshaping what is computationally possible. However, their potential to advance the probabilistic numerics paradigm - and thereby transform scientific computing - remains largely unexplored.

This PhD project will investigate how generative models can be harnessed to accelerate and improve numerical algorithms, with particular focus on:
1) Solvers for ODEs, PDEs, and SPDEs
2) Core computational tools such as basis decompositions and numerical differentiation
3) Applications in collaboration with scientific partners with numerical models, spanning climate science (e.g. atmospheric chemistry), systems biology, and computational fluid dynamics.

The successful candidate will gain expertise at the intersection of state-of-the-art machine learning, applied mathematics, and scientific computing. This training will provide an excellent foundation for a career in academia or industry at the highest international level.

Candidate profile:
Applicants should have a strong background in a discipline with significant mathematical or computational content (e.g. mathematics, computer science, physics, or computational life sciences).

Lead supervisor: Dr Henry Moss (henry.moss@lancaster.ac.uk)

Modern engineering challenges - such as designing advanced materials, optimising 3D structures, or developing complex systems -demand efficient methods to explore vast design spaces while respecting intricate feasibility constraints. Bayesian Optimisation (BO) is the current standard for solving high-cost design problems, but it struggles in these settings: while it can identify promising solutions, it often fails to incorporate the domain knowledge needed to ensure feasibility (e.g. manufacturability or physical validity).

The advent of domain-specific generative AI models, trained on design libraries and engineering datasets, offers a breakthrough. These models encapsulate decades of tacit expertise and can propose feasible, realistic designs on demand. Building on our recent work on generative AI–accelerated BO, this PhD project will develop next-generation methods to address multi-objective, multi-fidelity optimisation problems in engineering design.

The project will combine probabilistic machine learning, optimisation, and generative modelling, with opportunities to collaborate with international academic and industrial partners working on real-world engineering applications.

Candidate profile:
We welcome applicants with strong quantitative backgrounds in mathematics, computer science, engineering, or related fields. Experience in optimisation or machine learning is beneficial but not essential.