# Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements

arXiv: https://arxiv.org/abs/2410.10530 · HTML: https://mlsanity.com/paper/2410.10530/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 57 references that are in the map.
- First posted on arXiv 14 Oct 2024.

## Builds on (most cited first, 20 of 20)

- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [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)
- [Stable Implementation of Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.10106/) (2020)
- [Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systems](https://mlsanity.com/paper/2002.09301/) (2020)
- [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)
- [Fenrir: Physics-Enhanced Regression for Initial Value Problems](https://mlsanity.com/paper/2202.01287/) (2022)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [Pick-and-Mix Information Operators for Probabilistic ODE Solvers](https://mlsanity.com/paper/2110.10770/) (2021)
- [Probabilistic Exponential Integrators](https://mlsanity.com/paper/2305.14978/) (2023)
- [Confederated Modular Differential Equation APIs for Accelerated Algorithm Development and Benchmarking](https://mlsanity.com/paper/1807.06430/) (2018)
- [Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations](https://mlsanity.com/paper/2306.05566/) (2023)
- [Parallel-in-Time Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2310.01145/) (2023)
- [Numerically Robust Fixed-Point Smoothing Without State Augmentation](https://mlsanity.com/paper/2409.20004/) (2024)
- [Probabilistic ODE Solvers for Integration Error-Aware Numerical Optimal Control](https://mlsanity.com/paper/2401.17731/) (2024)
- [Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations](https://mlsanity.com/paper/2402.12231/) (2024)

## Abstract (arXiv)

Despite substantial progress in recent years, probabilistic solvers with adaptive step sizes can still not solve memory-demanding differential equations -- unless we care only about a single point in time (which is far too restrictive; we want the whole time series). Counterintuitively, the culprit is the adaptivity itself: Its unpredictable memory demands easily exceed our machine's capabilities, making our simulations fail unexpectedly and without warning. Still, dropping adaptivity would abandon years of progress, which can't be the answer. In this work, we solve this conundrum. We develop an adaptive probabilistic solver with fixed memory demands building on recent developments in robust state estimation. Switching to our method (i) eliminates memory issues for long time series, (ii) accelerates simulations by orders of magnitude through unlocking just-in-time compilation, and (iii) makes adaptive probabilistic solvers compatible with scientific computing in JAX.