Physically Inspired Machine Learning Models: Learning Differential Equations from Data

The University of Bath is inviting applications for the following PhD project in the Department of Computer Science commencing 30 September 2024.

The successful student will be part of the Centre for the Analysis of Motion, Entertainment Research and Applications (CAMERA) which performs world-leading multi-disciplinary research in Intelligent Visual and Interactive Technology. Funded by the EPSRC and the University of Bath, CAMERA exists to accelerate the impact of fundamental research being undertaken at the University in the Departments of Computer Science, Health and Psychology. The successful candidate will work closely work with the experts from CAMERA and potentially with collaborators from the University of Bristol and project partners associated with the MyWorld programme. The ambititious MyWorld project is funded by the UKRI Strength in Places fund bringing together 30 partners from Bristol and Bath’s creative technologies sector and world-leading academic institutions to create a unique cross-sector consortium.

Overview of this Project:

Simulation-based models form the backbone of predictive modelling in the natural sciences and engineering. Fundamentally, a scientific model is a simplification of reality that enables the understanding and prediction of essential aspects of a system, either to forecast its future behaviour or to optimize its design according to prescribed requirements.

While machine Learning (ML) techniques have been explored to provide fast predictions, it has been found that black-box approaches often struggle to match the performance of traditional methods. In recent years, Physics-based ML approaches are a promising new direction, wherein physical knowledge, such as governing equations or symmetries, are integrated into the ML model to improve the accuracy and reliability of predictions.

In particular, partial differential equations (PDEs) see widespread use in sciences and engineering due to the fact many scientific phenomena can be described by the evolution and interaction of physical quantities over space and time. Prominent examples include (i) fluid mechanics, which has applications in domains ranging from mechanical and civil engineering, to geophysics and meteorology, and (ii) electromagnetism, which provides mathematical models for electric, optical, or radio technologies.

For the majority of these equations, solutions are analytically intractable, and obtaining accurate predictions necessitates falling back on numerical approximation schemes often with prohibitive computation costs. This has led to neural surrogates becoming an active research topic in an attempt to accelerate these simulations. While, they have demonstrated solving PDEs many orders of magnitude faster, the practical utility of training such surrogates is contingent on their ability to successfully model phenomena across different spatial and temporal scales, which is a notoriously hard problem.

What are you going to do?

We seek a PhD Candidate that will contribute to research at the intersection of traditional PDE-based and ML methods.

One opportunity in this space is to leverage the flexibility of ML techniques to learn fast approximate solutions to PDEs. A second frontier, with huge potential, is learning PDE-based models from observational data. For lumped parameter models (systems of coupled ordinary differential equations) this has already been demonstrated. However, for spatio-temporal systems, which could be modelled by PDEs, only very few examples exist.

An alternate avenue of interest relates to training neural surrogates. While current methods model local and global relationships, they do not explicitly take into account the relationship between different fields and their internal components, which are often correlated. Viewing the time evolution of such correlated fields through the lens of multi-vector fields, as described by Clifford algebra, could overcome these limitations.

For further details, please see https://www.ndfcampbell.org/opportunities/

Candidate Requirements:

Applicants should hold, as a minimum, a First Class or good Upper Second Class UK Honours degree (or the equivalent) in a relevant subject.

Applicants without a Master's level degree in computer vision, computer graphics, machine learning, applied mathematics, physics, or a strongly correlated field, would have to provide strong justification (and evidence) that they would be able to handle the maths and programming necessary to complete a PhD in this field.

Programming experience is a particular advantage, specifically proficiency in numerical Python / C++ or similar. All of the techniques we use build on Linear Algebra and it would be desirable for the candidate to have experience in applied mathematics / numerical methods.

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

Enquiries and Applications:

Informal enquiries are welcomed and should be directed to Prof Neill Campbell (email: [Email Address Removed]).

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

When completing the application form, please:

1. In the Funding your studies section, select ‘University of Bath LURS’ as the studentship for which you are applying and quote ‘MyWorld LURS’ in the further information box.
2. In the Your PhD project section, quote the project title and lead supervisor in the appropriate boxes.

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

NOTE: Applications may close earlier than the advertised deadline if a suitable candidate is found. We therefore recommend that you contact the lead supervisor prior to applying and submit your formal application as early as possible.

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.