# Structure-Preserving Learning Using Gaussian Processes and Variational Integrators

arXiv: https://arxiv.org/abs/2112.05451 · HTML: https://mlsanity.com/paper/2112.05451/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 30 references that are in the map.
- First posted on arXiv 10 Dec 2021.

## Builds on (most cited first, 19 of 19)

- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control](https://mlsanity.com/paper/2007.01926/) (2020)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [Feedback Linearization based on Gaussian Processes with event-triggered Online Learning](https://mlsanity.com/paper/1911.06565/) (2019)
- [Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applications](https://mlsanity.com/paper/2109.02606/) (2021)
- [Learning Constrained Dynamics with Gauss Principle adhering Gaussian Processes](https://mlsanity.com/paper/2004.11238/) (2020)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [Variational Integrator Graph Networks for Learning Energy Conserving Dynamical Systems](https://mlsanity.com/paper/2004.13688/) (2020)
- [How Training Data Impacts Performance in Learning-based Control](https://mlsanity.com/paper/2005.12062/) (2020)
- [A Linear-Time Variational Integrator for Multibody Systems](https://mlsanity.com/paper/1609.02898/) (2016)
- [Efficient Computation of Higher-Order Variational Integrators in Robotic Simulation and Trajectory Optimization](https://mlsanity.com/paper/1904.12756/) (2019)
- [Linear-Time Variational Integrators in Maximal Coordinates](https://mlsanity.com/paper/2002.11245/) (2020)
- [Linear-Quadratic Optimal Control in Maximal Coordinates](https://mlsanity.com/paper/2010.05886/) (2020)
- [Linear-Time Contact and Friction Dynamics in Maximal Coordinates using Variational Integrators](https://mlsanity.com/paper/2109.07262/) (2021)

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

- [Exact Inference for Continuous-Time Gaussian Process Dynamics](https://mlsanity.com/paper/2309.02351/) (2023)
- [Data-efficient Kernel Methods for Learning Hamiltonian Systems](https://mlsanity.com/paper/2509.17154/) (2025)
- [Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations](https://mlsanity.com/paper/2605.06246/) (2026)

## Abstract (arXiv)

Gaussian process regression is increasingly applied for learning unknown dynamical systems. In particular, the implicit quantification of the uncertainty of the learned model makes it a promising approach for safety-critical applications. When using Gaussian process regression to learn unknown systems, a commonly considered approach consists of learning the residual dynamics after applying some generic discretization technique, which might however disregard properties of the underlying physical system. Variational integrators are a less common yet promising approach to discretization, as they retain physical properties of the underlying system, such as energy conservation and satisfaction of explicit kinematic constraints. In this work, we present a novel structure-preserving learning-based modelling approach that combines a variational integrator for the nominal dynamics of a mechanical system and learning residual dynamics with Gaussian process regression. We extend our approach to systems with known kinematic constraints and provide formal bounds on the prediction uncertainty. The simulative evaluation of the proposed method shows desirable energy conservation properties in accordance with general theoretical results and demonstrates exact constraint satisfaction for constrained dynamical systems.