# Learning ODE Models with Qualitative Structure Using Gaussian Processes

arXiv: https://arxiv.org/abs/2011.05364 · HTML: https://mlsanity.com/paper/2011.05364/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Nov 2020.

## Built on by (most cited first, 9 of 9)

- [Variational multiple shooting for Bayesian ODEs with Gaussian processes](https://mlsanity.com/paper/2106.10905/) (2021)
- [Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior](https://mlsanity.com/paper/2305.09017/) (2023)
- [Symmetry-Informed Governing Equation Discovery](https://mlsanity.com/paper/2405.16756/) (2024)
- [Hamiltonian Neural Networks with Automatic Symmetry Detection](https://mlsanity.com/paper/2301.07928/) (2023)
- [Learning Energy Conserving Dynamics Efficiently with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2303.01925/) (2023)
- [Uncertainty and Structure in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2305.13290/) (2023)
- [The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Models](https://mlsanity.com/paper/2211.11103/) (2022)
- [Learning solutions of parameterized stiff ODEs using Gaussian processes](https://mlsanity.com/paper/2511.05990/) (2025)
- [MAGI-X: Manifold-Constrained Gaussian Process Inference for Unknown System Dynamics](https://mlsanity.com/paper/2105.12894/) (2021)

## Abstract (arXiv)

Recent advances in learning techniques have enabled the modelling of dynamical systems for scientific and engineering applications directly from data. However, in many contexts explicit data collection is expensive and learning algorithms must be data-efficient to be feasible. This suggests using additional qualitative information about the system, which is often available from prior experiments or domain knowledge. We propose an approach to learning a vector field of differential equations using sparse Gaussian Processes that allows us to combine data and additional structural information, like Lie Group symmetries and fixed points. We show that this combination improves extrapolation performance and long-term behaviour significantly, while also reducing the computational cost.