# Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis

arXiv: https://arxiv.org/abs/2305.13879 · HTML: https://mlsanity.com/paper/2305.13879/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 89 references that are in the map.
- First posted on arXiv 23 May 2023.

## Builds on (most cited first, 21 of 21)

- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [A Case Study Competition Among Methods for Analyzing Large Spatial Data](https://mlsanity.com/paper/1710.05013/) (2017)
- [A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion](https://mlsanity.com/paper/1308.1313/) (2013)
- [A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables](https://mlsanity.com/paper/2006.01789/) (2020)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Physics-Informed CoKriging: A Gaussian-Process-Regression-Based Multifidelity Method for Data-Model Convergence](https://mlsanity.com/paper/1811.09757/) (2018)
- [Gaussian Process Regression constrained by Boundary Value Problems](https://mlsanity.com/paper/2012.11857/) (2020)
- [The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running](https://mlsanity.com/paper/2111.01084/) (2021)
- [Non-separable Spatio-temporal Graph Kernels via SPDEs](https://mlsanity.com/paper/2111.08524/) (2021)
- [APIK: Active Physics-Informed Kriging Model with Partial Differential Equations](https://mlsanity.com/paper/2012.11798/) (2020)
- [The rational SPDE approach for Gaussian random fields with general smoothness](https://mlsanity.com/paper/1711.04333/) (2017)
- [Mitigating the Influence of the Boundary on PDE-based Covariance Operators](https://mlsanity.com/paper/1610.05280/) (2016)
- [Statistical Finite Elements via Langevin Dynamics](https://mlsanity.com/paper/2110.11131/) (2021)
- [Stochastic Modeling of Inhomogeneities in the Aortic Wall and Uncertainty Quantification using a Bayesian Encoder-Decoder Surrogate](https://mlsanity.com/paper/2202.10244/) (2022)

## Built on by (most cited first, 1 of 1)

- [Variational Bayesian surrogate modelling with application to robust design optimisation](https://mlsanity.com/paper/2404.14857/) (2024)

## Abstract (arXiv)

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning. Today's prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest. The SPDE representation is especially appealing for engineering applications which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. We use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on complex geometries. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The respective mean vector and precision matrix and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with Poisson and thin-shell examples.