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

arXiv: https://arxiv.org/abs/2109.04757 · HTML: https://mlsanity.com/paper/2109.04757/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 47 references that are in the map.
- First posted on arXiv 10 Sep 2021.

## Builds on (most cited first, 6 of 6)

- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Data Assimilation: A Mathematical Introduction](https://mlsanity.com/paper/1506.07825/) (2015)
- [Pseudo-Marginal Bayesian Inference for Gaussian Processes](https://mlsanity.com/paper/1310.0740/) (2013)
- [Parallel inversion of huge covariance matrices](https://mlsanity.com/paper/1312.1869/) (2013)

## Built on by (most cited first, 3 of 3)

- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Phi-DVAE: Physics-Informed Dynamical Variational Autoencoders for Unstructured Data Assimilation](https://mlsanity.com/paper/2209.15609/) (2022)
- [Statistical Finite Elements via Langevin Dynamics](https://mlsanity.com/paper/2110.11131/) (2021)

## Abstract (arXiv)

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.