mlsanityData to 14 Oct 2026

Probabilistic ODE Solutions in Millions of Dimensions

arXiv 2110.11812 ↗First posted 22 Oct 2021stat.ML cs.LG cs.NAcited by 13

Not on the map. Why?

Abstract (arXiv)

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving {high-dimensional} ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems -- most importantly, the solution of discretised {partial} differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.

Key facts. Cited by 13 papers in the map (data through 14 Oct 2026). Builds on 12 of its 36 references that are in the map. First posted on arXiv 22 Oct 2021.

Builds on

12 of 36 references.

References · 12

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
  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. 2014Probabilistic ODE Solvers with Runge-Kutta Meanscited by 22
  6. 2020Calibrated Adaptive Probabilistic ODE Solverscited by 21
  7. 2021A Probabilistic State Space Model for Joint Inference from Differential Equations and Datacited by 21
  8. 2020Bayesian ODE Solvers: The Maximum A Posteriori Estimatecited by 19
  9. 2020Stable Implementation of Probabilistic ODE Solverscited by 19
  10. 2018Convergence Rates of Gaussian ODE Filterscited by 14
  11. 2020Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systemscited by 14
  12. 2020Deep Gaussian Markov Random Fieldscited by 8

Built on by

Cited by year (2026 to 14 Oct)

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

13 papers.

Sort:

  1. 2022Fenrir: Physics-Enhanced Regression for Initial Value Problemscited by 11
  2. 2022A Kernel Approach for PDE Discovery and Operator Learningcited by 7
  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. 2024Scaling up Probabilistic PDE Simulators with Structured Volumetric Informationcited by 2
  7. 2024Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equationscited by 1
  8. 2025ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fieldscited by 1
  9. 2024Beyond Flatland: A Geometric Take on Matching Methods for Treatment Effect Estimationcited by 0
  10. 2024Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirementscited by 0
  11. 2026Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equationscited by 0
  12. 2026Scalable Bayesian Inference for Nonlinear Conservation Lawscited by 0
1 more
  1. 2026Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEscited by 0