Research

Combining theory, modelling, data and computation to unlock the next generation of deep learning.

Maths4DL has become a leader in the growing field of scientific machine learning, which lies on the intersection of mathematics, computer science and various application areas.

We have successfully identified areas where machine learning is not performing well on scientific problems (challenging the existing paradigm), finding out how to make it perform better, increasing its accuracy and speeding up its training. The methods developed have been applied to challenging data sets and problems arising both in academia and industry. In this rapidly changing field, our research has evolved to focus on three key areas.

Machine learning for Inverse problems

The state-of-the-art in inverse problems has rapidly shifted to data-driven techniques, most of which do not have much underlying mathematical theory. We are investigating cutting edge techniques that get the most out of data while preserving important mathematical structures.

To this end, we are investigating better training algorithms (eg bilevel learning [1]), training regimes (eg unsupervised learning [2]), learned forward operators [3], regularizer architectures (eg with generative models [3]) and reconstruction methods [4,5,6].

Machine learning for differential equations

Machine learning is increasingly being applied to solve problems formulated in terms of differential equations, either directly approximating the solution or by constructing the corresponding solution operators. There is currently a lack of a general convergence theory for such methods, coupled with significant issues with the training algorithms.

To address this issue, we are studying both of these areas, together with the linked problem of representing neural networks in terms of deterministic, and stochastic, ODEs. We are investigating structure preserving ML [1], neural operators [2], PINNS; comparison with FE, approximation and convergence [3,4], dynamics of neural networks and neural ODES [5], mesh generation using ML and PDEs [6], score based diffusion for generative AI [7].

Applications

We focus on applications where deep learning is interconnected with physical modes and we are interested in reliable and accurate predictions.

Examples are plentiful, and we are currently investigating applications in environmental sciences (eg weather prediction) [1], biomedical imaging (eg magnetic resonance imaging [2] and computed tomography [3]), material sciences (eg ptychography [4]) and energy optimisation and uncertainty (eg power and telecommunications networks).

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