The Department of Mathematics and Mathematical Statistics is opening a PhD position in mathematics, within the WASP AI program, focusing on geometric deep learning. The position covers four years of third-cycle studies, including participation in research and third-cycle courses. The last day to apply is 10 June 2024.

This recruitment is part of a more significant expansion of the research group at the department investigating mathematical foundations of artificial intelligence. The group covers a diverse range of topics in modern machine learning research, including geometric deep learning, non-convex optimization problems and federated learning. In addition, this project is deeply connected with the geometry group at the department, a recently formed and inspiring research environment that is also expanding.

Project description and tasks:
Deep learning has enjoyed tremendous success on an impressive number of complex problems. However, the fundamental mathematical understanding of deep learning models is still incomplete, presenting exciting research problems spanning areas such as differential geometry, numerical analysis, and dynamical systems. Neural ordinary differential equations (NODEs) mark a recent advance in geometric deep learning, the pursuit to incorporate symmetries and non-Euclidean structures in machine learning using geometrical principles. NODEs describe the dynamics of information propagating through neural networks in the limit of infinite depth using ordinary differential equations (ODEs) on manifolds and offer several appealing properties.

The dynamical systems in NODE models are constrained, however, in that the intrinsic nature of the dimension of a manifold fixes the dimension of their state vector. This limitation precludes the use of certain architectural elements, like the encoder-decoder structure used in autoencoders and sequence-to-sequence prediction, and applications where the dimensionality of the state space changes dynamically, like quantum mechanical systems interacting with classical external fields where quantization effects cause freeze-out of degrees of freedom.

To remedy these limitations, the overarching goal of this project is to accommodate variable dimension dynamics in geometric deep learning by extending NODEs from manifolds to M-polyfolds, a generalization of manifolds where the number of local coordinates is allowed to vary smoothly. This requires the development of a comprehensive geometric framework for flows and integral curves on M-Polyfolds and a theory of group actions compatible with the M-polyfold structure.

The project is a part of the AI-Math track within Wallenberg AI, Autonomous Systems and Software Program (WASP). The PhD student will participate in the WASP graduate school.

Umeå offers excellent working and living conditions. The city is young and located right next to a large river. It is surrounded by forests and lakes and lies close by the sea. In the vicinity there are plenty of opportunities for both indoor and outdoor activities.

For further information and instructions on how to apply, see:

https://umu.varbi.com/en/what:job/jobID:723175/