  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 the posterior mean and
variance, paving the way for further integration of physical and probabilistic
approaches to complex systems.
