# The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Models

arXiv: https://arxiv.org/abs/2211.11103 · HTML: https://mlsanity.com/paper/2211.11103/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 23 references that are in the map.
- First posted on arXiv 20 Nov 2022.

## Builds on (most cited first, 12 of 12)

- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Data-Efficient Reinforcement Learning with Probabilistic Model Predictive Control](https://mlsanity.com/paper/1706.06491/) (2017)
- [Actively Learning Gaussian Process Dynamics](https://mlsanity.com/paper/1911.09946/) (2019)
- [Probabilistic ODE Solvers with Runge-Kutta Means](https://mlsanity.com/paper/1406.2582/) (2014)
- [Overcoming Mean-Field Approximations in Recurrent Gaussian Process Models](https://mlsanity.com/paper/1906.05828/) (2019)
- [Variational multiple shooting for Bayesian ODEs with Gaussian processes](https://mlsanity.com/paper/2106.10905/) (2021)
- [On Simulation and Trajectory Prediction with Gaussian Process Dynamics](https://mlsanity.com/paper/1912.10900/) (2019)
- [Learning ODE Models with Qualitative Structure Using Gaussian Processes](https://mlsanity.com/paper/2011.05364/) (2020)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [Linearized Gaussian Processes for Fast Data-driven Model Predictive Control](https://mlsanity.com/paper/1812.10579/) (2018)

## Built on by (most cited first, 2 of 2)

- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2503.04684/) (2025)

## Abstract (arXiv)

Computing the distribution of trajectories from a Gaussian Process model of a dynamical system is an important challenge in utilizing such models. Motivated by the computational cost of sampling-based approaches, we consider approximations of the model's output and trajectory distribution. We show that previous work on uncertainty propagation, focussed on discrete state-space models, incorrectly included an independence assumption between subsequent states of the predicted trajectories. Expanding these ideas to continuous ordinary differential equation models, we illustrate the implications of this assumption and propose a novel piecewise linear approximation of Gaussian Processes to mitigate them.