Partial differential equations for machine learning

The University of Bath is inviting applications for the following PhD project commencing in October 2022.

Partial differential equations are an essential tool for the mathematical modelling of biological, socio-economic and physical processes, and play a crucial role in the rapid advance of these disciplines. In recent years, large data sets have become available in almost all areas of engineering, science and technology. This motivates to analyse these data sets and develop novel pathways between partial differential equations, big data sets and machine learning.

Many machine learning methods are currently mainly driven by the outcomes. Mathematical theory, required to justify the methods used in practise, lags behind the rapid progress in applications and raises questions in terms of reliability of results, particularly for life-threatening decisions in healthcare. Since data is expected to play an even bigger role in our daily lives in the future, it is of great importance to develop systematic theoretical foundations and computational tools for the analysis of high-dimensional data sets and their integration into mathematical models.

This project uses partial differential equations for machine learning and is split into two main objectives: the analysis of big data sets with applications in semi-supervised learning in the first part and the integration of large data sets into mathematical models using machine learning techniques in the second part.

In terms of semi-supervised learning applications, important mathematical questions include finding structures within data sets and developing mathematical methods to identify them efficiently. Since the data points of a given data set are related to each other in a particular way, the data points can be represented as nodes of a graph and their relations can be expressed through edges, leading to the representation of the data as a graph or a network. This data representation motivates to study differential equations on graphs which is a new, promising field arising in many applications in high-dimensional data analysis. This part of the project involves studying differential equations on graphs by both establishing analytical foundations and developing computational methodologies.

The new insights into the data sets can be used for the seamless integration of big data sets into mathematical models. Based on machine learning techniques, the integration of large data sets into mathematical models will improve calibration and predictions. This part of the project investigates the mathematical formulation of the problem both analytically and numerically with the aim of developing efficient computational schemes.

One of the major application areas of this project will be in climate modelling. Climate change is reshaping our world, but more accurate predictions are required to face the coming changes. Bath has very close links with the Met Office and this project will take advantage of these through visits at the Met Office and in particular interactions with the Met Office’s data assimilation group to improve the exploitation of satellite data in numerical weather prediction. The methods developed in this project can be used for data analysis of satellite data. In a later stage, the Met Office’s forecasting procedures, involving techniques from data assimilation and numerical analysis of differential equations, will be applied to combine data and numerical weather predictions. 

Project keywords: partial differential equations, numerical analysis, mathematical analysis, data analysis.

Candidate Requirements:

Applicants should hold, or expect to receive, a First Class or good Upper Second Class Honours degree (or the equivalent). A master’s level qualification would also be advantageous.

Non-UK applicants must meet our English language entry requirement.

Enquiries and Applications:

Informal enquiries are welcomed and should be directed to Dr Lisa Maria Kreusser on email address [Email Address Removed].

Formal applications should be made via the University of Bath’s online application form for a PhD in Mathematics.

More information about applying for a PhD at Bath may be found on our website.

Funding Eligibility:

To be eligible for funding, you must qualify as a Home student. The eligibility criteria for Home fee status are detailed and too complex to be summarised here in full; however, as a general guide, the following applicants will normally qualify subject to meeting residency requirements: UK nationals (living in the UK or EEA/Switzerland), Irish nationals (living in the UK or EEA/Switzerland), those with Indefinite Leave to Remain and EU nationals with pre-settled or settled status in the UK under the EU Settlement Scheme). This is not intended to be an exhaustive list. Additional information may be found on our fee status guidance webpage, on the GOV.UK website and on the UKCISA website.

Exceptional Overseas students (e.g. with a UK Master’s Distinction or international equivalent and relevant research experience), who are interested in this project, should contact the lead supervisor in the first instance to discuss the possibility of applying for supplementary funding.

Equality, Diversity and Inclusion:

We value a diverse research environment and aim to be an inclusive university, where difference is celebrated and respected. We welcome and encourage applications from under-represented groups.

If you have circumstances that you feel we should be aware of that have affected your educational attainment, then please feel free to tell us about it in your application form. The best way to do this is a short paragraph at the end of your personal statement.