PhD position in Mathematics focusing on geometric deep learning
1 month ago
Project description
and tasksDeep 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.
Qualifications
The doctoral student will be admitted to the third-cycle programme in Mathematics. To fulfil the general entry requirements, the applicant must have qualifications equivalent to a completed degree at second-cycle level or completed course requirements of at least 240 ECTS credits, including at least 60 ECTS credits at second-cycle level. To fulfil the specific entry requirements to be admitted for studies at third-cycle level in mathematics, the applicant must have completed at least 60 ECTS credits within the field of mathematics, of which at least 15 ECTS credits shall have been acquired at second-cycle level. Applicants who have acquired largely equivalent skills in some other system, either within Sweden or abroad, are also eligible.
Good programming skills and a good knowledge of the English language, both written and spoken, are required. Documented knowledge and experience in differential geometry, differential equations, and machine learning are meritorious but not required.
The doctoral student is expected to take on an active role in developing the research project and in departmental work. Therefore, they are expected to have excellent communication and collaboration skills. They should have a scientific mindset, the ability to work independently and be structured, flexible and solution-oriented. Above all, the doctoral student should be analytical, creative, and committed to continuously developing their skills and contributing to the mathematical foundations of machine learning.
The assessment of the applicants is based on their qualifications and ability to benefit from the doctoral education they will receive.
About the employment
The position is intended to result in a doctoral degree. The main task of the doctoral student is to pursue their third-cycle studies, including active participation in research and third-cycle courses, and participate in the WASP graduate school. The duties may include teaching or other departmental work, although duties of this kind may not comprise more than 20 per cent of a full-time post. The employment is for a fixed term of four years full-time or up to five years when teaching part-time. Salary is set according to the salary ladder for PhD positions at Umeå University. Employment commences in January 2025 or by agreement.
The graduate school within WASP is dedicated to provide the skills needed to analyze, develop, and contribute to the interdisciplinary area of artificial intelligence, autonomous systems and software. Through an ambitious program with research visits, partner universities, and visiting lecturers, the graduate school actively supports forming a strong multi-disciplinary and international professional network between PhD-students, researchers and industry.
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