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

TL;DR

This paper introduces a low-rank approximation technique for statFEM, enabling scalable Bayesian finite element modeling that effectively combines physical laws with data, even in high-dimensional systems.

Contribution

It presents a novel low-rank covariance approximation method that enhances the scalability of statFEM for complex, high-dimensional problems.

Findings

Accurately reconstructs data-generating processes with minimal information loss.

Maintains posterior mean and variance quality in high-dimensional reaction-diffusion problems.

Demonstrates computational efficiency and scalability improvements.

Abstract

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.