# Gaussian processes for Bayesian inverse problems associated with linear partial differential equations

arXiv: https://arxiv.org/abs/2307.08343 · HTML: https://mlsanity.com/paper/2307.08343/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 44 references that are in the map.
- First posted on arXiv 17 Jul 2023.

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

- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priors](https://mlsanity.com/paper/2201.06463/) (2022)
- [Iterative Updating of Model Error for Bayesian Inversion](https://mlsanity.com/paper/1707.04246/) (2017)
- [Introduction To Gaussian Process Regression In Bayesian Inverse Problems, With New ResultsOn Experimental Design For Weighted Error Measures](https://mlsanity.com/paper/2302.04518/) (2023)
- [Random forward models and log-likelihoods in Bayesian inverse problems](https://mlsanity.com/paper/1712.05717/) (2017)

## Built on by (most cited first, 9 of 9)

- [Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.08501/) (2025)
- [Gaussian Process Regression for Inverse Problems in Linear PDEs](https://mlsanity.com/paper/2502.04276/) (2025)
- [An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation](https://mlsanity.com/paper/2503.00213/) (2025)
- [Sampling as Bandits: Evaluation-Efficient Design for Black-Box Densities](https://mlsanity.com/paper/2509.01437/) (2025)
- [Physics-informed Gaussian Process Regression in Solving Eigenvalue Problem of Linear Operators](https://mlsanity.com/paper/2601.06462/) (2026)
- [Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs](https://mlsanity.com/paper/2603.03922/) (2026)
- [Target localization, identification and sensing using latent symmetries](https://mlsanity.com/paper/2606.01421/) (2026)
- [Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations](https://mlsanity.com/paper/2606.20417/) (2026)
- [A unified perspective of Gaussian process approximation for differential equations](https://mlsanity.com/paper/2607.06292/) (2026)

## Abstract (arXiv)

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.