Artificial intelligence at mathematical sciences

The Department of Mathematical Sciences has a core role in these developments. Our researchers are working on different topics within AI, for example mathematical foundations, improved algorithms, new applications, and societal impacts.

This WASP AI/Math project aims to shed light on the mathematical structure of deep neural networks using techniques and insights from a variety of different fields in mathematics and physics, including quantum mechanics, information theory, differential geometry, group theory and gauge theory.
Despite the overwhelming success of deep neural networks we are still at a loss for explaining exactly how deep learning works, and why it works so well. What is the mathematical framework underlying deep learning? One promising direction is to consider symmetries as an underlying design principle for network architectures. This can be implemented by constructing deep neural networks on a group G that acts transitively on the input data. This is directly relevant for instance in the case of spherical signals where G is a rotation group.
Even more generally, it is natural to consider the question of how to train neural networks in the case of “non-flat” data. Relevant applications include omnidirectional computer vision, biomedicine, and climate observations, just to mention a few situations where data is naturally “curved”. Mathematically, this calls for developing a theory of deep learning on manifolds, or even more exotic structures, like graphs or algebraic varieties. A special class consists of homogeneous spaces G/H, where H is a subgroup. The project aims to develop the mathematical framework for convolutional neural networks which are globally equivariant with respect to G and locally with respect to H. The formalism is closely similar to gauge theory in physics and exploring this connection is a potential side project.
Daniel Persson is PI for the project. Christoffer Petersson, Deep Learning Research Engineer at Zenseact and Adjunct Associate Professor at Mathematical Sciences, is industrial supervisor in the project. Fredrik Ohlsson, Associate Professor at Department of Mathematics and Mathematical Statistics at Umeå University, is an external researcher in the project.
<h4>Papers</h4>
<a href="https://arxiv.org/abs/2105.05400" title="Link to publication">Homogeneous vector bundles and G-equivariant convolutional neural networks</a>, Jimmy Aronsson 
<a href="https://arxiv.org/abs/2105.13926" title="Link to publication">Geometric Deep Learning and Equivariant Neural Networks</a>, Jan E. Gerken, Jimmy Aronsson, Oscar Carlsson, Hampus Linander, Fredrik Ohlsson, Christoffer Petersson, Daniel Persson

Jan Gerken
Fredrik Olsson, Umeå University

Olof Zetterqvist is PhD student in a WASP AI/Math project supervised by Johan Jonasson and Rebecka Jörnsten. The project is about practical and theoretical aspects of sensitivity analysis for neural networks. They are also looking at topic models and convergence analysis for MCMC.

To ensure that AI becomes a net positive for humanity we need to work out how to navigate the technology landscape in such a way as to reap its positive potential while avoiding the worst risks – this is the key motivation behind Olle Häggström’s interest in AI safety and AI governance. A popular summary of some of the reasons leading towards this motivation is offered in <a href="https://www.youtube.com/watch?v=HFJULkDwXnM" title="Link to Youtube" data-anchor="?v=HFJULkDwXnM">a talk he gave in September 2018</a> (in Swedish). Initially his work on AI safety was mainly on how to <a href="https://drops.dagstuhl.de/opus/volltexte/2018/8898/pdf/LIPIcs-AofA-2018-5.pdf">handle the extreme scenario</a> in which an AI with superhuman general intelligence is created, but more recently he has shifted some of his attention towards more down-to-earth settings, such the social effects of robotisation, and risks from chatbots and other powerful tools for political propaganda, as well as from AI systems for autonomous weapons.

The project is about understanding partially observed high-dimensional point processes, with an aim at diagnosis of side effects of diabetes and chemo therapy.
Spatial point pattern data are more and more often collected in large areas, where one may have hundreds of different species of plants followed over time, or collected for diagnostic purposes, where several (point pattern) samples are taken from several subjects having different conditions (belonging to different groups). Questions related to the data are also getting more complex. Instead of finding a suitable model for one point pattern, model parameters need to be estimated, for example, from data having the hierarchical structure above, i.e. data from different individuals and groups. Furthermore, methods to handle measurement errors and partially observed data are not widely studied in the point process literature. Methods from machine learning, such as clustering and variational Bayes methods, can become very useful, but they need first to be adjusted for spatial point processes.

​In this WASP AI/Math project PhD student Filip Wikman is studying the relation between Deep Neural Networks (DNN), Partial Differential Equations, and Stochastic Analysis. Supervisor for the project is Annika Lang. The aim is to obtain a better understanding of the underlying structure of DNN, and to construct more efficient training algorithms.

In this project we advance the understanding of deep neural networks through the investigation of stochastic continuous-depth neural networks. These can be thought of as deep neural networks (DNN) composed of infinitely many stochastic layers, where each single layer only brings about a gradual change to the output of the preceding layers. We will analyse such stochastic continuous-depth neural networks using tools from stochastic calculus and Bayesian statistics. From that, we will derive practically relevant and novel training algorithms for stochastic DNNs with the aim to capture the uncertainty associated with the predictions of the network.
Go to <a href="https://research.chalmers.se/project/9214" title="Link to Chalmers' research database">Chalmers' research database</a> to find more information and publications related to the project.

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Mathematical Sciences

AI@MS – Artificial intelligence at mathematical sciences

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Geometric deep learning and equivariant neural networks

Sensitivity analysis for neural networks

AI – a threat or an opportunity?

Understanding partially observed point processes

Machine Learning and SPDE

Stochastic continuous-depth neural networks

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