# Learning solutions of parameterized stiff ODEs using Gaussian processes

arXiv: https://arxiv.org/abs/2511.05990 · HTML: https://mlsanity.com/paper/2511.05990/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 35 references that are in the map.
- First posted on arXiv 8 Nov 2025.

## Builds on (most cited first, 11 of 11)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Input Warping for Bayesian Optimization of Non-stationary Functions](https://mlsanity.com/paper/1402.0929/) (2014)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems](https://mlsanity.com/paper/2010.14685/) (2020)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methods](https://mlsanity.com/paper/2410.05592/) (2024)
- [Learning ODE Models with Qualitative Structure Using Gaussian Processes](https://mlsanity.com/paper/2011.05364/) (2020)
- [How many samples are needed to train a deep neural network?](https://mlsanity.com/paper/2405.16696/) (2024)
- [Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs](https://mlsanity.com/paper/2208.12045/) (2022)

## Built on by (most cited first, 1 of 1)

- [Trajectory-Optimized Time Reparameterization for Learning-Compatible Reduced-Order Modeling of Stiff Dynamical Systems](https://mlsanity.com/paper/2603.16583/) (2026)

## Abstract (arXiv)

Stiff ordinary differential equations (ODEs) play an important role in many scientific and engineering applications. Often, the dependence of the solution of the ODE on additional parameters is of interest, e.g. when dealing with uncertainty quantification or design optimization. Directly studying this dependence can quickly become too computationally expensive, such that cheaper surrogate models approximating the solution are of interest. One popular class of surrogate models are Gaussian processes (GPs). They perform well when approximating stationary functions, functions which have a similar level of variation along any given parameter direction, however solutions to stiff ODEs are often characterized by a mixture of regions of rapid and slow variation along the time axis and when dealing with such nonstationary functions, GP performance frequently degrades drastically. We therefore aim to reparameterize stiff ODE solutions based on the available data, to make them appear more stationary and hence recover good GP performance. This approach comes with minimal computational overhead and requires no internal changes to the GP implementation, as it can be seen as a separate preprocessing step. We illustrate the achieved benefits using multiple examples.