mlsanityData to 14 Oct 2026

Bayesian Numerical Methods for Nonlinear Partial Differential Equations

arXiv 2104.12587 ↗First posted 22 Apr 2021math.NA cs.NA stat.COcited by 15

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Abstract (arXiv)

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of the PDE is identified using novel theoretical analysis of the sample path properties of Mat'ern processes, which may be of independent interest.

Key facts. Cited by 15 papers in the map (data through 14 Oct 2026). Builds on 24 of its 67 references that are in the map. First posted on arXiv 22 Apr 2021.

Builds on

24 of 67 references.

References · 24

  1. 2018Gaussian Processes and Kernel Methods: A Review on Connections and Equivalencescited by 192
  2. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  3. 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
  4. 2015Probabilistic Numerics and Uncertainty in Computationscited by 60
  5. 2017Bayesian Probabilistic Numerical Methodscited by 51
  6. 2017Linearly constrained Gaussian processescited by 42
  7. 2016A probabilistic model for the numerical solution of initial value problemscited by 32
  8. 2018Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspectivecited by 31
  9. 2017Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexitycited by 24
  10. 2014Probabilistic ODE Solvers with Runge-Kutta Meanscited by 22
  11. 2020Calibrated Adaptive Probabilistic ODE Solverscited by 21
  12. 2020Bayesian ODE Solvers: The Maximum A Posteriori Estimatecited by 19
12 more
  1. 2019A Modern Retrospective on Probabilistic Numericscited by 18
  2. 2021High-Dimensional Gaussian Process Inference with Derivativescited by 16
  3. 2020Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functionscited by 15
  4. 2018Convergence Rates of Gaussian ODE Filterscited by 14
  5. 2016Active Uncertainty Calibration in Bayesian ODE Solverscited by 9
  6. 2020APIK: Active Physics-Informed Kriging Model with Partial Differential Equationscited by 9
  7. 2016Probabilistic Linear Multistep Methodscited by 5
  8. 2018Implicit Probabilistic Integrators for ODEscited by 5
  9. 2021Small Sample Spaces for Gaussian Processescited by 5
  10. 2016Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problemscited by 3
  11. 2020Probabilistic Gradients for Fast Calibration of Differential Equation Modelscited by 3
  12. 2019A Role for Symmetry in the Bayesian Solution of Differential Equationscited by 2

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 2; 2022: 2; 2023: 3; 2024: 4; 2025: 3; 2026: 1.

15 papers.

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  1. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 41
  2. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  3. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  4. 2021Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equationscited by 13
  5. 2023Sample Path Regularity of Gaussian Processes from the Covariance Kernelcited by 11
  6. 2024Toward Efficient Kernel-Based Solvers for Nonlinear PDEscited by 7
  7. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  8. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
  9. 2024Scaling up Probabilistic PDE Simulators with Structured Volumetric Informationcited by 2
  10. 2022Dynamic Bayesian Learning for Spatiotemporal Mechanistic Modelscited by 1
  11. 2024Probabilistic Numeric SMC Sampling for Bayesian Nonlinear System Identification in Continuous Timecited by 1
  12. 2025Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fieldscited by 1
3 more
  1. 2021Black Box Probabilistic Numericscited by 0
  2. 2025Bayesian Nonlinear PDE Inference via Gaussian Process Collocation with Application to the Richards Equationcited by 0
  3. 2026Adaptive multi-resolution Gaussian processes: Scalable exact inference with naturally data-sparse covariance matricescited by 0