Machine learning and its applications have grown explosively in the last few years. However, the success of machine learning is still poorly understood from a theoretical point of view. Current state-of-the-art machine learning techniques are known to lack robustness and often come without theoretical guarantees, which questions the reliability of the results. This becomes particularly concerning for safety-critical applications such as life-threatening decisions in medicine.
In contrast, differential equations are of great importance in scientific computing and are an essential tool for modelling in science and engineering. In addition, they are based on rigorous and well understood scientific principles. With the emergence of deep neural networks in modern machine learning, this motivates us to make use of their complementary strengths: the modelling power and interpretability of differential equations, and the approximation and generalisation power of neural networks.
This project will unlock the next generation of transparent machine learning by developing new algorithms and theoretical foundations for hybrid approaches at the interface of machine learning and differential equations. There are two main directions on how synergies between differential equations and machine learning can result in transparent AI. Firstly, differential equations can be used for addressing challenges in machine learning. For instance, diffusion models which emerged as the new state-of-the-art generative models are based on differential equations, and the study of such models will give new insights, particularly in terms of the analysis and design of new computational tools. Secondly, machine learning can be used to learn dynamics from data or solve differential equations. An example for this is the numerical solution of high-dimensional problems or nonsmooth phenomena such turbulences or high frequencies which are nearly impossible to resolve using standard methods.
Unlike most state-of-the-art machine learning approaches, the methods developed in this project will not be fully black-box methods. By incorporating classical numerical techniques, these hybrid computational methods will not only be fast, but also provide a degree of transparency that is lacking in current approaches. Additionally, incorporating physical constraints will also allow the algorithm to converge to the correct solution faster. The algorithm development will be complemented by the mathematical analysis of the methods which will establish theoretical guarantees on the convergence of the algorithm and the conservation of desirable properties of the solution.
This project is associated with the UKRI Centre for Doctoral Training (CDT) in Accountable, Responsible and Transparent AI (ART-AI). We value people from different life experiences with a passion for research. The CDT's mission is to graduate diverse specialists with perspectives who can go out in the world and make a difference.
Applicants should hold, or expect to receive, a First or Upper Second Class Honours degree in Mathematics or disciplines with significant mathematical components such as Computer Science, Physics and Chemistry. Experience in programming (in any language) is desirable.
Informal enquiries about the project should be directed to Dr Kreusser.
Formal applications should be accompanied by a research proposal and made via the University of Bath’s online application form. Enquiries about the application process should be sent to firstname.lastname@example.org.
Start date: 2 October 2023.