Self-started summer project conducted under the supervision of a lecturer at Imperial College. I was invited to present the project as a finalist at a school competition and Imperial's Centre for Complexity Science.
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Many natural phenomena display properties or behaviours more than the mere aggregation of their parts. Humans, for instance, are capable of language, cognition and intricate social behaviours, none of which are properties of individual cells. Similarly, each cell’s functionality arises from the interactions between molecules, even though none possess the cell’s capabilities independently. This pattern, where macroscopic properties arise from interactions between microscopic components, termed ’emergence’, is a hallmark of complex systems. Emergence creates layers of abstraction within a system, where each behaves according to its own physical laws. Formal theories of emergence have already been introduced using information theory, such as in [5]. The contribution of this paper is a novel method of identifying emergence using machine learning. By approximating the dynamics of a complex system at different spatiotemporal scales, I confirm numerically that these layers of abstraction exist, and that the dynamics of each can be learned by a data-driven approach. I evaluate this method using the Classical XY model, a lattice model of statistical mechanics relevant to phenomena such as the melting of crystals, magnetism and superconductivity, as an example. At the microscopic scale, the model consists of a collection of spins on a lattice that can point in any direction in the plane, which operate according to the dynamics of equation 4. At the macroscopic scale, the model is characterised by emergent structures termed ’vortices’ and ’anti-vortices’, which describe topological flaws where groups of spins make a 2π rotation either clockwise or anticlockwise, that follow Coulomb dynamics. To this end, I propose a dual pathway approach to predicting the trajectories of spins and vortices using graph neural networks. First, I trained a model to predict spin dynamics, from which the vortices could be extracted. Second, I trained a model that bypasses spins, instead directly predicting vortex movements. By drawing parallels to commutativity diagrams, I demonstrate that both pathways converge to accurate vortex predictions, even over extended rollouts.