PhD student position at the Department of Mathematics and Mathematical Statistics, Umeå University.
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.