The Mathematics of Deep Learning

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences


Machine learning (ML), in particular Deep Learning (DL) is one of the fastest growing areas of modern science and technology, which has potentially enormous and transformative impact on all areas of our life. The applications of DL embrace many disciplines such as (bio-)medical sciences, computer vision, the physical sciences, the social sciences, speech recognition, gaming, music and finance. DL based algorithms are now used to play chess and GO at the highest level, diagnose illness, drive cars, recruit staff and even make legal judgements. The possible applications in the future are almost unlimited. Perhaps DL methods will be used in the future to predict the weather and climate, of even human behaviour. However, alongside this explosive growth has been a concern that there is a lack of explainability behind DL and the way that DL based algorithms make their decisions. This leads to a lack of trustworthiness in the use of the algorithms. A reason for this is that the huge successes of deep learning is not well understood, the results are mysterious, and there is a lack of a clear link between the data training DL algorithms (which is often vague and unstructured) and the decisions made by these algorithms.

Part of the reason for this is that DL has advanced so fast, that there is a lack of understanding of its foundations. According to the leading computer scientist Ali Rahimi at NIPS 2017: 'We say things like "machine learning is the new electricity". I'd like to offer another analogy. Machine learning has become alchemy!'
Indeed, despite the roots of ML lying in mathematics, statistics and computer science there currently is hardly any rigorous mathematical theory for the setup, training and application performance of deep neural networks.

We urgently need the opportunity to change machine learning from alchemy into science. This programme grant aims to rise to this challenge, and, by doing so, to unlock the future potential of artificial intelligence. It aims to put deep learning onto a firm mathematical basis, and will combine theory, modelling, data, computation to unlock the next generation of deep learning.

The grant will comprise an interlocked set of work packages aimed to address both the theoretical development of DL (so that it becomes explainable) and the algorithmic development (so that it becomes trustworthy). These will then be linked to the development of DL in a number of key application areas including image processing, partial differential equations and environmental problems. For example we will explore the question of whether it is possible to use DL based algorithms to forecast the weather and climate faster and more accurately than the existing physics based algorithms.

The investigators on the grant will be doing both theoretical investigations and will work with end-users of DL in many application areas. Mindful that policy makers are trying to address the many issues raised by DL, the investigators will also reach out to them through a series of workshops and conferences. The results of the work will also be presented to the public at science festivals and other open events.


10 25 50
Description Policy workshop
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Description Statistical modelling and minimization of energy use in 5G networks
Amount £104,903 (GBP)
Funding ID 230033 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 09/2023 
End 09/2026
Description BT 
Organisation BT Group
Country United Kingdom 
Sector Private 
PI Contribution Projects with BT, talks and several workshops
Collaborator Contribution Working on projects and particiapation in workshops
Impact Ongoing
Start Year 2022
Description ESO Partnership 
Organisation National Grid ESO
Country United Kingdom 
Sector Public 
PI Contribution Significant collaboration on projects linking machine learning to electricity supply
Collaborator Contribution Supply of data
Impact MSc thesis
Start Year 2022
Description Met Office 
Organisation Meteorological Office UK
Country United Kingdom 
Sector Academic/University 
Start Year 2004
Description British Library 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Public/other audiences
Results and Impact A presentation from the Maths4DL team at a big science fair at the British library in Septermber 2022
Year(s) Of Engagement Activity 2022
Description Popular article 
Form Of Engagement Activity A magazine, newsletter or online publication
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Schools
Results and Impact The outcomes of the grant have been written up as a series of popular articles for the PLUS mathematics magasine. These have been several articles which have appeared in 2022 (and also 2023) aimed at a school audience.
Year(s) Of Engagement Activity 2022
Description RI Masterclass 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Schools
Results and Impact A Royal Institution masterclass presentation to KS3
Year(s) Of Engagement Activity 2022