PhD Studentships: Connections between Numerical Analysis of Differential Equations and Machine Learning

This is a fully funded 3.5 year PhD project; funding will cover fees and provide a tax free stipend set at the UKRI rate (£19,237 for 2024/25). Funding is available for UK students and EU students with settles status.

Up to two funded PhD projects are available in the Department of Mathematics at the University of Manchester on Connections between Numerical Analysis of Differential Equations and Machine Learning. Candidates must have an MSc at distinction level (or equivalent) in applied mathematics, numerical analysis or similar and be eligible for UK fee status. Candidates should have a solid background in numerical analysis and in applying numerical techniques to solve differential equations. Experience in programming in a language such as python, MATLAB or other is also essential. Prior experience in machine learning or other AI techniques is not essential but is certainly desirable. Students will be affiliated with the EPSRC-funded Probabilistic AI Hub, a five-year collaboration between the Universities of Manchester, Lancaster, Edinburgh, Cambridge, Bristol and Warwick on the Mathematical and Computational Foundations of AI.

Machine learning (ML) and AI methods are becoming increasingly popular as data-driven approaches to building surrogates for physical models or as models for complex data. While they offer more flexibility in how data can be incorporated into the approximation process, standard approaches often ignore, or are not able to enforce important structural information. For example, in fluid flow modelling, we often require velocity approximations to be mass-conserving. In structural engineering problems, approximations may need to have specific spatial structures due to geometric constraints. While there has been some research that attempts to incorporate this information into an AI model, these have had limited success for complex and nonlinear problems compared to bespoke adaptive numerical methods derived using rigorous numerical analysis arguments. Finding novel ways to fuse structural and other domain-specific information, where it exists, with data holds the promise to not only produce better AI models that are easier to fit but also more robust approximations that have smaller generalisation error. In this project, students will develop connections between numerical analysis of, and numerical methods for differential equations (deterministic, parametric, stochastic) and the design and analysis of novel AI methods. Project topics include, but are not limited to:

• Physics-reinforced ML methods
• Developing hybrid classical and AI-informed solvers for PDEs/parametric PDEs/stochastic PDEs
• Designing better neural network architectures that mimic structure-preserving numerical schemes for differential equations
• Using ML techniques to learn parameters that optimise performance of deterministic and randomised numerical methods
• Learning PDEs from data using variational and Bayesian techniques

Candidates must have an MSc at distinction level (or equivalent) in applied mathematics, numerical analysis or similar and be eligible for UK fee status.

We strongly encourage interested applications to contact Professor Catherine Powell before applying (Catherine.Powell@manchester.ac.uk).

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