I currently have an open, fully-funded PhD position (starting January 2024 or September 2024) at the University of Manchester. Please find an overview of potential research topics below and contact me, if you are interested.
Machine learning with partial differential equations
Machine learning and artificial intelligence play a major part in our everyday life. Self-driving cars, automatic diagnoses from medical images, face recognition, or fraud detection, all profit especially from the universal applicability of deep neural networks. Their use in safety critical applications, however, is problematic: no interpretability, missing mathematical guarantees for network or learning process, and no quantification of the uncertainties in the neural network output.
Recently, models that are based on partial differential equations (PDEs) have gained popularity in machine learning. In a classification problem, for instance, a PDE is constructed whose solution correctly classifies the training data and gives a suitable model to classify unlabelled feature vectors. In practice, feature vectors tend to be high dimensional and the natural space on which they live tends to have a complicated geometry. Therefore, partial differential equations on graphs are particularly suitable and popular. The resulting models are interpretable, mathematically well-understood, and uncertainty quantification is possible. In addition, they can be employed in a semi-supervised fashion, making them highly applicable in small data settings.
I am interested in various mathematical, statistical, and computational aspects of PDE-based machine learning. Many of those aspects translate easily into PhD projects; examples are
- Efficient algorithms for p-Laplacian-based regression and clustering
- Bayesian identification of graphs from flow data
- PDEs on random graphs
- Deeply learned PDEs in data science
Depending on the project, applicants should be familiar with at least one of: (a) numerical analysis and numerical linear algebra; (b) probability theory and statistics; (c) machine learning and deep learning.
If you are interested in a postdoctoral position, still contact me: there are several postdoctoral fellowship schemes that may be appropriate for us to apply jointly, e.g., the Newton International Fellowship or the Leverhulme Early Career Fellowship.