Fibration symmetries in biological and artificial neural networks

Networked interactions are crucial in shaping collective phenomena in complex dynamical systems [1]. In particular, synchronisation can be exhibited as a global state [2] in which all units follow the same trajectory or via clustered states where the system splits into subsets of synchronised units. In the latter phenomenon, known as cluster synchronisation [3], the symmetries inherent to the network structure of connections play a key role in determining the composition of the clusters [4].  Symmetry, on the other hand, is the bedrock of theoretical physics, over which our understanding of nature has been constructed, from classical mechanics to the standard model of particle physics [5]. Symmetries are synonymous with symmetry groups, which are global and often too restrictive to adequately capture the local processes in networked systems.  Inspired by the concept of Grothendieck's fibrations in category theory [6], a generalized theory of hierarchical symmetries spanning from fibrations [7] to coverings [8] can be constructed. This extends the strict symmetry groups of physics into local and less stringent forms. Symmetry has also emerged as a crucial concept for artificial neural networks, where it serves as an important inductive bias that enhances performance.  Along with their broken symmetries, these local symmetries describe how neural networks work during inference and learning, acting as organizing principles for biological networks as well [9-10]. 

 

 

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[10] Gili T et al. 2024, https://doi.org/10.21203/rs.3.rs-4409330/v1.