mlsanityData to 14 Oct 2026

Pick-and-Mix Information Operators for Probabilistic ODE Solvers

arXiv 2110.10770 ↗First posted 20 Oct 2021stat.ML cs.LG cs.NAcited by 6

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Abstract (arXiv)

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 14 of its 46 references that are in the map. First posted on arXiv 20 Oct 2021.

Builds on

14 of 46 references.

References · 14

  1. 2015Probabilistic Numerics and Uncertainty in Computationscited by 60
  2. 2017Bayesian Probabilistic Numerical Methodscited by 51
  3. 2016A probabilistic model for the numerical solution of initial value problemscited by 32
  4. 2018Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspectivecited by 31
  5. 2020Calibrated Adaptive Probabilistic ODE Solverscited by 21
  6. 2020Bayesian ODE Solvers: The Maximum A Posteriori Estimatecited by 19
  7. 2020Stable Implementation of Probabilistic ODE Solverscited by 19
  8. 2019A Modern Retrospective on Probabilistic Numericscited by 18
  9. 2019ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systemscited by 17
  10. 2018Convergence Rates of Gaussian ODE Filterscited by 14
  11. 2015On Computational Complexity Reduction Methods for Kalman Filter Extensionscited by 5
  12. 2018Implicit Probabilistic Integrators for ODEscited by 5
2 more
  1. 2015Partitioned Update Kalman Filtercited by 1
  2. 2016Kullback-Leibler Divergence Approach to Partitioned Update Kalman Filtercited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 3; 2024: 1; 2025: 0; 2026: 1.

6 papers.

  1. 2022Fenrir: Physics-Enhanced Regression for Initial Value Problemscited by 11
  2. 2023Probabilistic Exponential Integratorscited by 4
  3. 2023Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equationscited by 3
  4. 2023Parallel-in-Time Probabilistic Numerical ODE Solverscited by 3
  5. 2024Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirementscited by 0
  6. 2026Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEscited by 0