# Parallel-in-Time Probabilistic Numerical ODE Solvers

arXiv: https://arxiv.org/abs/2310.01145 · HTML: https://mlsanity.com/paper/2310.01145/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 65 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 25 of 25)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Temporal Parallelization of Bayesian Smoothers](https://mlsanity.com/paper/1905.13002/) (2019)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Calibrated Adaptive Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.08202/) (2020)
- [A Probabilistic State Space Model for Joint Inference from Differential Equations and Data](https://mlsanity.com/paper/2103.10153/) (2021)
- [Bayesian ODE Solvers: The Maximum A Posteriori Estimate](https://mlsanity.com/paper/2004.00623/) (2020)
- [Stable Implementation of Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.10106/) (2020)
- [A Modern Retrospective on Probabilistic Numerics](https://mlsanity.com/paper/1901.04457/) (2019)
- [Convergence Rates of Gaussian ODE Filters](https://mlsanity.com/paper/1807.09737/) (2018)
- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
- [Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2110.11847/) (2021)
- [Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics](https://mlsanity.com/paper/1306.0308/) (2013)
- [Fenrir: Physics-Enhanced Regression for Initial Value Problems](https://mlsanity.com/paper/2202.01287/) (2022)
- [Active Uncertainty Calibration in Bayesian ODE Solvers](https://mlsanity.com/paper/1605.03364/) (2016)
- [Parallel Iterated Extended and Sigma-point Kalman Smoothers](https://mlsanity.com/paper/2102.00514/) (2021)
- [Pick-and-Mix Information Operators for Probabilistic ODE Solvers](https://mlsanity.com/paper/2110.10770/) (2021)
- [Parallel square-root statistical linear regression for inference in nonlinear state space models](https://mlsanity.com/paper/2207.00426/) (2022)
- [Probabilistic Exponential Integrators](https://mlsanity.com/paper/2305.14978/) (2023)
- [Iterated Extended Kalman Smoother-based Variable Splitting for L1-Regularized State Estimation](https://mlsanity.com/paper/1903.08605/) (2019)

## Built on by (most cited first, 3 of 3)

- [Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements](https://mlsanity.com/paper/2410.10530/) (2024)
- [Recursive Entropic Variational Inference for Nonlinear State-Space Models](https://mlsanity.com/paper/2511.15409/) (2025)
- [Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs](https://mlsanity.com/paper/2606.08203/) (2026)

## Abstract (arXiv)

Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic flexibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this flexibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the span cost from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.