# Exact Inference for Continuous-Time Gaussian Process Dynamics

arXiv: https://arxiv.org/abs/2309.02351 · HTML: https://mlsanity.com/paper/2309.02351/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 5 Sep 2023.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Probabilistic Recurrent State-Space Models](https://mlsanity.com/paper/1801.10395/) (2018)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [ResNet After All? Neural ODEs and Their Numerical Solution](https://mlsanity.com/paper/2007.15386/) (2020)
- [Fast Gaussian Process Based Gradient Matching for Parameter Identification in Systems of Nonlinear ODEs](https://mlsanity.com/paper/1804.04378/) (2018)
- [Discovery of Dynamics Using Linear Multistep Methods](https://mlsanity.com/paper/1912.12728/) (2019)
- [Symplectic Gaussian Process Regression of Hamiltonian Flow Maps](https://mlsanity.com/paper/2009.05569/) (2020)
- [On Simulation and Trajectory Prediction with Gaussian Process Dynamics](https://mlsanity.com/paper/1912.10900/) (2019)
- [Probabilistic Linear Multistep Methods](https://mlsanity.com/paper/1610.08417/) (2016)
- [Learning nonparametric differential equations with operator-valued kernels and gradient matching](https://mlsanity.com/paper/1411.5172/) (2014)
- [Structure-Preserving Learning Using Gaussian Processes and Variational Integrators](https://mlsanity.com/paper/2112.05451/) (2021)

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

- [System-Aware Neural ODE Processes for Few-Shot Bayesian Optimization](https://mlsanity.com/paper/2406.02352/) (2024)
- [Learning Passive Continuous-Time Dynamics with Multistep Port-Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2510.00384/) (2025)
- [A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems](https://mlsanity.com/paper/2606.13063/) (2026)

## Abstract (arXiv)

Physical systems can often be described via a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. This can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. Higher-order numerical integrators provide the necessary tools to address this problem by discretizing the dynamics function with arbitrary accuracy. Many higher-order integrators require dynamics evaluations at intermediate time steps making exact GP inference intractable. In previous work, this problem is often tackled by approximating the GP posterior with variational inference. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to make direct inference tractable, we propose to leverage multistep and Taylor integrators. We demonstrate how to derive flexible inference schemes for these types of integrators. Further, we derive tailored sampling schemes that allow to draw consistent dynamics functions from the learned posterior. This is crucial to sample consistent predictions from the dynamics model. We demonstrate empirically and theoretically that our approach yields an accurate representation of the continuous-time system.