Allen completed his MSc in Mathematics and Physics at the University of Bath. During his summers he worked at l’Institut Laue-Langevin, Grenoble, France looking at the scattering patterns of neutrons diffracted through materials like benzene, graphene and ice. His university projects have been varied and include simulating critical fluids and calibrating an antenna array – but he doesn’t know what area of mathematics or which application to commit to at the moment. He hopes to find a fun problem during his first year at SAMBa he can attack throughout the remainder of his PhD. Outside of mathematics, Allen has done a bit of work as a ski instructor, likes to play sports, and hang out with friends.

Research project title:
Echo State Networks and their application to dynamical systems

Supervisor(s):
Jonathan Dawes

Project description:
Allen is studying how well a particular recurrent neural network architecture called the Echo State Network (ESN) can approximate dynamical systems, predicting their future behaviour as well as inferring their topological features. Allen hopes to use ideas from Takens’ Embedding Theorem to prove that an ESN trained on a time series of low dimensional observations of a high dimensional dynamical system can learn the topology of the high dimensional system. Having learned the topology to some level of precision, the ideas from the Universal Approximation Theorem could be deployed to prove that a sufficiently large ESN trained on sufficiently many data can predict the future dynamics of a system arbitrarily well. Numerical experiments will also provide some intuition about how well practical ESNs perform on example dynamical systems like the Lorenz, or Mackey-Glass systems.