# The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

arXiv: https://arxiv.org/abs/2111.01084 · HTML: https://mlsanity.com/paper/2111.01084/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2021.

## Built on by (most cited first, 15 of 15)

- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Gaussian Process Regression under Computational and Epistemic Misspecification](https://mlsanity.com/paper/2312.09225/) (2023)
- [Uncertainty Estimation on Graphs with Structure Informed Stochastic Partial Differential Equations](https://mlsanity.com/paper/2506.06907/) (2025)
- [Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis](https://mlsanity.com/paper/2305.13879/) (2023)
- [Implementation and Analysis of GPU Algorithms for Vecchia Approximation](https://mlsanity.com/paper/2407.02740/) (2024)
- [Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields](https://mlsanity.com/paper/2503.08343/) (2025)
- [LatticeVision: Image to Image Networks for Modeling Non-Stationary Spatial Data](https://mlsanity.com/paper/2505.09803/) (2025)
- [Discrete Gaussian Vector Fields On Meshes](https://mlsanity.com/paper/2507.20024/) (2025)
- [BALLAST: Bayesian Active Learning with Look-ahead Amendment for Sea-drifter Trajectories under Spatio-Temporal Vector Fields](https://mlsanity.com/paper/2509.26005/) (2025)
- [Exploiting Hankel-Toeplitz Structures for Fast Computation of Kernel Precision Matrices](https://mlsanity.com/paper/2408.02346/) (2024)
- [Linear cost and exponentially convergent approximation of Gaussian Mat'ern processes on intervals](https://mlsanity.com/paper/2410.13000/) (2024)
- [On the Effect of Regularization on Nonparametric Mean-Variance Regression](https://mlsanity.com/paper/2511.22004/) (2025)
- [Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds](https://mlsanity.com/paper/2512.10407/) (2025)
- [Time-adaptive infinite-dimensional Gaussian process regression on manifolds](https://mlsanity.com/paper/2603.21144/) (2026)
- [Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds](https://mlsanity.com/paper/2606.13827/) (2026)

## Abstract (arXiv)

Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Mat'ern covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space-time fields on arbitrary manifolds, and practical computational considerations.