# Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations

arXiv: https://arxiv.org/abs/2605.06246 · HTML: https://mlsanity.com/paper/2605.06246/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 50 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 28 of 28)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Symplectic Gaussian Process Regression of Hamiltonian Flow Maps](https://mlsanity.com/paper/2009.05569/) (2020)
- [Physics-informed Neural Networks to Model and Control Robots: a Theoretical and Experimental Investigation](https://mlsanity.com/paper/2305.05375/) (2023)
- [Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior](https://mlsanity.com/paper/2305.09017/) (2023)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Generalized Lagrangian Neural Networks](https://mlsanity.com/paper/2401.03728/) (2024)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [A Structure-Preserving Kernel Method for Learning Hamiltonian Systems](https://mlsanity.com/paper/2403.10070/) (2024)
- [Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems](https://mlsanity.com/paper/2211.16006/) (2022)
- [Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics](https://mlsanity.com/paper/2505.20370/) (2025)
- [Learning Energy Conserving Dynamics Efficiently with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2303.01925/) (2023)
- [Machine Learning with Physics Knowledge for Prediction: A Survey](https://mlsanity.com/paper/2408.09840/) (2024)
- [Physically Consistent Learning of Conservative Lagrangian Systems with Gaussian Processes](https://mlsanity.com/paper/2206.12272/) (2022)
- [Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery](https://mlsanity.com/paper/2211.10830/) (2022)
- [A Black-Box Physics-Informed Estimator based on Gaussian Process Regression for Robot Inverse Dynamics Identification](https://mlsanity.com/paper/2310.06585/) (2023)
- [Structure-Preserving Learning Using Gaussian Processes and Variational Integrators](https://mlsanity.com/paper/2112.05451/) (2021)
- [Generalizable and Fast Surrogates: Model Predictive Control of Articulated Soft Robots using Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.01916/) (2025)
- [Physically Consistent Modeling & Identification of Nonlinear Friction with Dissipative Gaussian Processes](https://mlsanity.com/paper/2405.17199/) (2024)
- [Newtonian and Lagrangian Neural Networks: A Comparison Towards Efficient Inverse Dynamics Identification](https://mlsanity.com/paper/2506.17994/) (2025)
- [Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2511.05330/) (2025)
- [Simultaneous State Estimation and Online Model Learning in a Soft Robotic System](https://mlsanity.com/paper/2602.14092/) (2026)

## Abstract (arXiv)

In this paper, we propose Lagrangian Gaussian Processes (LGPs) for probabilistic and data-efficient learning of dynamics via discrete forced Euler-Lagrange equations. Importantly, the geometric structure of the Lagrange-d'Alembert principle, which governs the motion of dynamical systems, is preserved by construction in the absence of external forces. This allows learning physically consistent models that overcome erroneous drift in the system's energy, thereby providing stable long-term predictions. At the core of our approach lie linear operators for Gaussian process conditioning, constructed from discrete forced Euler-Lagrange equations and variational discretization schemes. Thereby and unlike prior work, the method enables learning dynamics from discrete position snapshots, i.e., without access to a system's velocities or momenta. This is particularly relevant for a large class of practical scenarios where only position measurements are available, for instance, in motion capture or visual servoing applications. We demonstrate the data-efficiency and generalization capabilities of the LGPs in various synthetic and real-world case studies, including a real-world soft robot with hysteresis. The experimental results underscore that the LGPs learn physically consistent dynamics with uncertainty quantification solely from sparse positional data and enable stable long-term predictions.