mlsanityData to 7 Oct 2026

Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations

arXiv 2110.11847 ↗First posted 22 Oct 2021math.NA cs.NA stat.MLcited by 13

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Abstract (arXiv)

This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 22 of its 59 references that are in the map. First posted on arXiv 22 Oct 2021.

Builds on

22 of 59 references.

References · 22

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  4. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
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  10. 2020Calibrated Adaptive Probabilistic ODE Solverscited by 21
  11. 2021A Probabilistic State Space Model for Joint Inference from Differential Equations and Datacited by 21
  12. 2017Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problemscited by 19
10 more
  1. 2020Bayesian ODE Solvers: The Maximum A Posteriori Estimatecited by 19
  2. 2020Stable Implementation of Probabilistic ODE Solverscited by 19
  3. 2015Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Gamescited by 15
  4. 2021Bayesian Numerical Methods for Nonlinear Partial Differential Equationscited by 15
  5. 2018Convergence Rates of Gaussian ODE Filterscited by 14
  6. 2020Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systemscited by 14
  7. 2013Probabilistic Solutions to Differential Equations and their Application to Riemannian Statisticscited by 11
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Citing papers per year: 2022: 3; 2023: 3; 2024: 4; 2025: 3.

13 papers.

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  1. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
  2. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  3. 2024Physics-Informed Variational State-Space Gaussian Processescited by 7
  4. 2022Sobolev Spaces, Kernels and Discrepancies over Hyperspherescited by 5
  5. 2023Probabilistic Exponential Integratorscited by 4
  6. 2023Parallel-in-Time Probabilistic Numerical ODE Solverscited by 3
  7. 2023Simulation-Based Inference of Surface Accumulation and Basal Melt Rates of an Antarctic Ice Shelf from Isochronal Layerscited by 2
  8. 2024Scaling up Probabilistic PDE Simulators with Structured Volumetric Informationcited by 2
  9. 2024Practical Bayesian Algorithm Execution via Posterior Samplingcited by 2
  10. 2025Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fieldscited by 1
  11. 2025StFT: Spatio-temporal Fourier Transformer for Long-term Dynamics Predictioncited by 1
  12. 2024Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirementscited by 0
1 more
  1. 2025Recursive Entropic Variational Inference for Nonlinear State-Space Modelscited by 0