# Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes

arXiv: https://arxiv.org/abs/2511.05330 · HTML: https://mlsanity.com/paper/2511.05330/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 41 references that are in the map.
- First posted on arXiv 7 Nov 2025.

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

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [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)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming](https://mlsanity.com/paper/2004.11408/) (2020)
- [Particle Gibbs with Ancestor Sampling](https://mlsanity.com/paper/1401.0604/) (2014)
- [A flexible state space model for learning nonlinear dynamical systems](https://mlsanity.com/paper/1603.05486/) (2016)
- [Symplectic Gaussian Process Regression of Hamiltonian Flow Maps](https://mlsanity.com/paper/2009.05569/) (2020)
- [Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior](https://mlsanity.com/paper/2305.09017/) (2023)
- [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)
- [Structured learning of rigid-body dynamics: A survey and unified view from a robotics perspective](https://mlsanity.com/paper/2012.06250/) (2020)
- [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)
- [Gaussian processes for dynamics learning in model predictive control](https://mlsanity.com/paper/2502.02310/) (2025)
- [Parameter elimination in particle Gibbs sampling](https://mlsanity.com/paper/1910.14145/) (2019)
- [Physically Consistent Learning of Conservative Lagrangian Systems with Gaussian Processes](https://mlsanity.com/paper/2206.12272/) (2022)
- [A Black-Box Physics-Informed Estimator based on Gaussian Process Regression for Robot Inverse Dynamics Identification](https://mlsanity.com/paper/2310.06585/) (2023)
- [On Foundation Models for Dynamical Systems from Purely Synthetic Data](https://mlsanity.com/paper/2412.00395/) (2024)
- [Amortized Bayesian Workflow](https://mlsanity.com/paper/2409.04332/) (2024)
- [Nonlinear System Identification: Learning while respecting physical models using a sequential Monte Carlo method](https://mlsanity.com/paper/2210.14684/) (2022)
- [Physically Consistent Modeling & Identification of Nonlinear Friction with Dissipative Gaussian Processes](https://mlsanity.com/paper/2405.17199/) (2024)
- [Bayesian Inference and Learning in Nonlinear Dynamical Systems: A Framework for Incorporating Explicit and Implicit Prior Knowledge](https://mlsanity.com/paper/2508.15345/) (2025)

## Built on by (most cited first, 1 of 1)

- [Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations](https://mlsanity.com/paper/2605.06246/) (2026)

## Abstract (arXiv)

Embedding non-restrictive prior knowledge, such as energy conservation laws, into learning methods is a key motive to construct physically consistent dynamics models from limited data, relevant for, e.g., model-based control. Recent work incorporates Hamiltonian dynamics into Gaussian Processes (GPs) to obtain uncertainty-quantifying, energy-consistent models, but these methods rely on -- rarely available -- velocity or momentum data. In this paper, we study dynamics learning using Hamiltonian GPs and focus on learning solely from input-output data, without relying on velocity or momentum measurements. Adopting a non-conservative formulation, energy exchange with the environment, e.g., through external forces or dissipation, can be captured. We provide a fully Bayesian scheme for estimating probability densities of unknown hidden states, GP hyperparameters, as well as structural hyperparameters, such as damping coefficients. The proposed method is evaluated in a nonlinear simulation case study and compared to a state-of-the-art approach that relies on momentum measurements.