[School of Natural Sciences PhD Scholarships] Hybrid Machine Learning & Finite Element Solvers for UQ in PDE Models

About the project

About the Project In forward uncertainty quantification (UQ), one aims to understand how uncertainty in model inputs affects predictions of quantities of interest associated with model solutions. Spatially varying uncertain inputs such as material coefficients and boundary conditions can be represented mathematically as random fields. Naive sampling methods then require repeated numerical solution of the physical model for different samples of the random inputs, drawn from an appropriate probability distribution. When the cost of solving the model numerically once to the desired level of accuracy is already problematic, such as when using a traditional finite element method (FEM) for PDE models, obtaining accurate uncertainty assessments becomes problematic. Over the last three decades, mathematicians and engineers have developed a range of sophisticated approaches for building surrogate models to tackle this issue, with rigorous error control possible for some PDE classes. However, machine learning (ML) techniques (eg physics-informed neural networks) are emerging as a disruptive technology, challenging traditional FEM-based approaches to solving PDEs and building surrogates for UQ studies. This project will explore hybrid numerical strategies for facilitating forward UQ in PDE models that combine the strengths of both emerging machine learning (ML) techniques and classical FE approximation. There is flexibility to adapt the project to the skills and interest of the student. Potential ideas include using AI/ML tools to generate samples of random inputs, to construct components of reduced basis methods that rely on finite element snapshots, and/or to learn components of solvers or preconditioners for discretised FE linear systems. The overall aim will be to develop hybrid solution strategies with accuracy guarantees and the project will involve both rigorous mathematical analysis as well as computational implementation. Suitable applicants must have a background up to MSc-level (or equivalent) in numerical analysis, including numerical solution of PDEs, as well as programming experience, for example in MATLAB or Python. This project is expected to start in September 2027. Before you apply: We strongly recommend that you contact the supervisors for this project before you apply. How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Applied Mathematics OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Applied Mathematics. Previous research experience in numerical solution of PDEs is desirable. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

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