mlsanityData to 7 Oct 2026

Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems

arXiv 1701.04006 ↗First posted 15 Jan 2017stat.ME cs.NA math.NAcited by 19

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Abstract (arXiv)

This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem with a nonlinear forward model.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). First posted on arXiv 15 Jan 2017.

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Citing papers per year: 2021: 3; 2022: 3; 2023: 4; 2024: 3; 2025: 5; 2026: 1.

19 papers.

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  1. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  2. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
  3. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  4. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  5. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  6. 2021Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equationscited by 13
  7. 2023Gaussian processes for Bayesian inverse problems associated with linear partial differential equationscited by 9
  8. 2024Physics-Informed Variational State-Space Gaussian Processescited by 7
  9. 2023Introduction To Gaussian Process Regression In Bayesian Inverse Problems, With New ResultsOn Experimental Design For Weighted Error Measurescited by 6
  10. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
  11. 2022Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equationscited by 4
  12. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
7 more
  1. 2024Scaling up Probabilistic PDE Simulators with Structured Volumetric Informationcited by 2
  2. 2025Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fieldscited by 1
  3. 2021Black Box Probabilistic Numericscited by 0
  4. 2025Are Statistical Methods Obsolete in the Era of Deep Learning? A Study of ODE Inverse Problemscited by 0
  5. 2025Scalable h-adaptive probabilistic solver for time-independent and time-dependent systemscited by 0
  6. 2025Bayesian Nonlinear PDE Inference via Gaussian Process Collocation with Application to the Richards Equationcited by 0
  7. 2026Operator-Informed Gaussian Processes for Complex Helmholtz Wavefields: From Synthetic Benchmarks to In Vivo Brain Elastographycited by 0