Backpropagation on Dynamical Networks

Dynamical networks are versatile models that can describe a variety of behaviours such as synchronisation and feedback. However, applying these models in real world contexts is difficult as prior information pertaining to the connectivity structure or local dynamics is often unknown and must be inferred from time series observations of network states. Additionally, the influence of coupling interactions between nodes further complicates the isolation of local node dynamics. Given the architectural similarities between dynamical networks and recurrent neural networks (RNN), we propose a network inference method based on the backpropagation through time (BPTT) algorithm commonly used to train recurrent neural networks. This method aims to simultaneously infer both the connectivity structure and local node dynamics purely from observation of node states. An approximation of local node dynamics is first constructed using a neural network. This is alternated with an adapted BPTT algorithm to regress corresponding network weights by minimising prediction errors of the dynamical network based on the previously constructed local models until convergence is achieved. This method was found to be succesful in identifying the connectivity structure for coupled networks of Lorenz, Chua and FitzHugh-Nagumo oscillators. Freerun prediction performance with the resulting local models and weights was found to be comparable to the true system with noisy initial conditions. The method is also extended to non-conventional network couplings such as asymmetric negative coupling.

Low-rank statistical finite elements for scalable model-data synthesis

Statistical learning additions to physically derived mathematical models are gaining traction in the literature. A recent approach has been to augment the underlying physics of the governing equations with data driven Bayesian statistical methodology. Coined statFEM, the method acknowledges a priori model misspecification, by embedding stochastic forcing within the governing equations. Upon receipt of additional data, the posterior distribution of the discretised finite element solution is updated using classical Bayesian filtering techniques. The resultant posterior jointly quantifies uncertainty associated with the ubiquitous problem of model misspecification and the data intended to represent the true process of interest. Despite this appeal, computational scalability is a challenge to statFEM's application to high-dimensional problems typically experienced in physical and industrial contexts. This article overcomes this hurdle by embedding a low-rank approximation of the underlying dense covariance matrix, obtained from the leading order modes of the full-rank alternative. Demonstrated on a series of reaction-diffusion problems of increasing dimension, using experimental and simulated data, the method reconstructs the sparsely observed data-generating processes with minimal loss of information, in both posterior mean and the variance, paving the way for further integration of physical and probabilistic approaches to complex systems.