Asghar, Solomon; (2025) Deep Learning for the Analysis and Generative Modelling of Anomalous Diffusion. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Manifesting in cosmic rays propagating across interstellar space, storks migrating to breed, and money flowing through financial markets, anomalous diffusion is exhibited by a diverse array of systems. Defined as diffusion with a mean square displacement that is not linear in time, this phenomenon has understandably garnered growing research interest across various disciplines. Despite this, modelling and analysing systems exhibiting anomalous diffusion remains challenging. In this thesis, I apply advances in deep learning to address both these issues. There is a shortage of efficient simulation frameworks for nonequilibrium diffusive dynamics. This hinders research into active matter, a promising class of inherently nonequilibrium materials that can exhibit anomalous diffusion. I introduce a novel framework, FlowRES, that uses unsupervised Normalising Flow neural networks to enhance Monte Carlo sampling of nonequilibrium diffusion by generating high-quality nonlocal Monte Carlo proposals. I validate FlowRES by sampling the transition path ensembles of equilibrium and non-equilibrium systems of Brownian particles exploring increasingly complex potential surfaces. Beyond eliminating requirements for prior data, FlowRES offers key advantages over established samplers: no collective variables need defining, its efficiency remains constant even as the events being sampled become increasingly rare, and it can handle systems with multiple routes between metastable states. Standard approaches for characterizing anomalous diffusion from individual trajectory measurements struggle in cases of practical interest: short or noisy trajectories, heterogeneous behaviour, and non-ergodic processes. I introduce a new U-Net based framework, U-AnD-ME, that allows for highly accurate ensemble level and trajectory level analysis of systems of particles exhibiting anomalous diffusion. The performance of this framework is evaluated using the results from the Anomalous Diffusion Challenge 2024, where quantitative comparison with a myriad of other methods showed U-AnD-ME to be the best available method for anomalous diffusion trajectory analysis at both the single trajectory and ensemble level.

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