mlsanityData to 7 Oct 2026

Fenrir: Physics-Enhanced Regression for Initial Value Problems

arXiv 2202.01287 ↗First posted 2 Feb 2022cs.LG stat.MLcited by 11

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Abstract (arXiv)

We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 20 of its 71 references that are in the map. First posted on arXiv 2 Feb 2022.

Builds on

20 of 71 references.

References · 20

  1. 2015Probabilistic Numerics and Uncertainty in Computationscited by 59
  2. 2017Bayesian Probabilistic Numerical Methodscited by 50
  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. 2021A Probabilistic State Space Model for Joint Inference from Differential Equations and Datacited by 21
  7. 2020Bayesian ODE Solvers: The Maximum A Posteriori Estimatecited by 19
  8. 2020Stable Implementation of Probabilistic ODE Solverscited by 19
  9. 2019A Modern Retrospective on Probabilistic Numericscited by 18
  10. 2018Fast Gaussian Process Based Gradient Matching for Parameter Identification in Systems of Nonlinear ODEscited by 17
  11. 2019ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systemscited by 17
  12. 2018Convergence Rates of Gaussian ODE Filterscited by 14
8 more
  1. 2020Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systemscited by 14
  2. 2021Probabilistic ODE Solutions in Millions of Dimensionscited by 13
  3. 2017Scalable Variational Inference for Dynamical Systemscited by 12
  4. 2021Pick-and-Mix Information Operators for Probabilistic ODE Solverscited by 6
  5. 2016Laplace based approximate posterior inference for differential equation modelscited by 5
  6. 2018Implicit Probabilistic Integrators for ODEscited by 5
  7. 2017Bayesian Filtering for ODEs with Bounded Derivativescited by 2
  8. 2019Estimation of ordinary differential equation models with discretization error quantificationcited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 6; 2024: 2; 2025: 2; 2026: 1.

11 papers.

  1. 2023Random Grid Neural Processes for Parametric Partial Differential Equationscited by 8
  2. 2023The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensionscited by 5
  3. 2023Probabilistic Exponential Integratorscited by 4
  4. 2023Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equationscited by 3
  5. 2023Parallel-in-Time Probabilistic Numerical ODE Solverscited by 3
  6. 2024Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equationscited by 1
  7. 2023Uncertainty Informed Optimal Resource Allocation with Gaussian Process based Bayesian Inferencecited by 0
  8. 2024Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirementscited by 0
  9. 2025Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solverscited by 0
  10. 2025Numerically robust Gaussian state estimation with singular observation noisecited by 0
  11. 2026Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEscited by 0