mlsanityData to 7 Oct 2026

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

arXiv 2307.08343 ↗First posted 17 Jul 2023stat.ML cs.LGcited by 9

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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.

Key facts. Cited by 9 papers in the map (data through 7 Oct 2026). Builds on 6 of its 44 references that are in the map. First posted on arXiv 17 Jul 2023.

Builds on

6 of 44 references.

References · 6

  1. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
  2. 2017Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problemscited by 19
  3. 2022Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priorscited by 9
  4. 2017Iterative Updating of Model Error for Bayesian Inversioncited by 6
  5. 2023Introduction To Gaussian Process Regression In Bayesian Inverse Problems, With New ResultsOn Experimental Design For Weighted Error Measurescited by 6
  6. 2017Random forward models and log-likelihoods in Bayesian inverse problemscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 4; 2026: 5.

9 papers.

  1. 2025Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networkscited by 2
  2. 2025Gaussian Process Regression for Inverse Problems in Linear PDEscited by 2
  3. 2025An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equationcited by 0
  4. 2025Sampling as Bandits: Evaluation-Efficient Design for Black-Box Densitiescited by 0
  5. 2026Physics-informed Gaussian Process Regression in Solving Eigenvalue Problem of Linear Operatorscited by 0
  6. 2026Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEscited by 0
  7. 2026Target localization, identification and sensing using latent symmetriescited by 0
  8. 2026Neural network surrogates with uncertainty quantification for inverse problems in partial differential equationscited by 0
  9. 2026A unified perspective of Gaussian process approximation for differential equationscited by 0