# Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control

arXiv: https://arxiv.org/abs/2106.12782 · HTML: https://mlsanity.com/paper/2106.12782/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 24 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models](https://mlsanity.com/paper/1805.12114/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Deep Lagrangian Networks for end-to-end learning of energy-based control for under-actuated systems](https://mlsanity.com/paper/1907.04489/) (2019)
- [Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics](https://mlsanity.com/paper/2005.14617/) (2020)
- [A General Framework for Structured Learning of Mechanical Systems](https://mlsanity.com/paper/1902.08705/) (2019)
- [AutoTune: Controller Tuning for High-Speed Flight](https://mlsanity.com/paper/2103.10698/) (2021)

## Built on by (most cited first, 30 of 38)

- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [Learning Articulated Rigid Body Dynamics with Lagrangian Graph Neural Network](https://mlsanity.com/paper/2209.11588/) (2022)
- [Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces](https://mlsanity.com/paper/2206.02660/) (2022)
- [Physics-Inspired Temporal Learning of Quadrotor Dynamics for Accurate Model Predictive Trajectory Tracking](https://mlsanity.com/paper/2206.03305/) (2022)
- [Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2212.00893/) (2022)
- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [Active Learning of Discrete-Time Dynamics for Uncertainty-Aware Model Predictive Control](https://mlsanity.com/paper/2210.12583/) (2022)
- [Distributed neural network control with dependability guarantees: a compositional port-Hamiltonian approach](https://mlsanity.com/paper/2112.09046/) (2021)
- [Pseudo-Hamiltonian neural networks for learning partial differential equations](https://mlsanity.com/paper/2304.14374/) (2023)
- [How to Learn and Generalize From Three Minutes of Data: Physics-Constrained and Uncertainty-Aware Neural Stochastic Differential Equations](https://mlsanity.com/paper/2306.06335/) (2023)
- [SkyJEPA: Learning Long-Horizon World Models for Zero-Shot Sim-to-Real Control of Quadrotors](https://mlsanity.com/paper/2606.23444/) (2026)
- [Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems](https://mlsanity.com/paper/2211.16006/) (2022)
- [Learning to Predict 3D Rotational Dynamics from Images of a Rigid Body with Unknown Mass Distribution](https://mlsanity.com/paper/2308.14666/) (2023)
- [A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics](https://mlsanity.com/paper/2410.18868/) (2024)
- [Adaptive Control of SE(3) Hamiltonian Dynamics with Learned Disturbance Features](https://mlsanity.com/paper/2109.09974/) (2021)
- [Guaranteed Conformance of Neurosymbolic Models to Natural Constraints](https://mlsanity.com/paper/2212.01346/) (2022)
- [Learning Interpretable Dynamics from Images of a Freely Rotating 3D Rigid Body](https://mlsanity.com/paper/2209.11355/) (2022)
- [Discovering interpretable Lagrangian of dynamical systems from data](https://mlsanity.com/paper/2302.04400/) (2023)
- [Deep Learning for Structure-Preserving Universal Stable Koopman-Inspired Embeddings for Nonlinear Canonical Hamiltonian Dynamics](https://mlsanity.com/paper/2308.13835/) (2023)
- [Forecasting Continuous Non-Conservative Dynamical Systems in SO(3)](https://mlsanity.com/paper/2508.07775/) (2025)
- [A Bayesian framework for discovering interpretable Lagrangian of dynamical systems from data](https://mlsanity.com/paper/2310.06241/) (2023)
- [Neural Port-Hamiltonian Models for Nonlinear Distributed Control: An Unconstrained Parametrization Approach](https://mlsanity.com/paper/2411.10096/) (2024)
- [Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach](https://mlsanity.com/paper/2509.24627/) (2025)
- [Data-driven discovery of non-Newtonian astronomy via learning non-Euclidean Hamiltonian](https://mlsanity.com/paper/2210.00090/) (2022)
- [Physics-Informed Kernel Embeddings: Integrating Prior System Knowledge with Data-Driven Control](https://mlsanity.com/paper/2301.03565/) (2023)
- [Pseudo-Hamiltonian system identification](https://mlsanity.com/paper/2305.06920/) (2023)
- [Learning Latent Dynamics via Invariant Decomposition and (Spatio-)Temporal Transformers](https://mlsanity.com/paper/2306.12077/) (2023)
- [Plug-and-Play Physics-informed Learning using Uncertainty Quantified Port-Hamiltonian Models](https://mlsanity.com/paper/2504.17966/) (2025)
- [Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equations](https://mlsanity.com/paper/2603.18331/) (2026)
- [Node Embedding from Hamiltonian Information Propagation in Graph Neural Networks](https://mlsanity.com/paper/2303.01030/) (2023)

## Abstract (arXiv)

Accurate models of robot dynamics are critical for safe and stable control and generalization to novel operational conditions. Hand-designed models, however, may be insufficiently accurate, even after careful parameter tuning. This motivates the use of machine learning techniques to approximate the robot dynamics over a training set of state-control trajectories. The dynamics of many robots, including ground, aerial, and underwater vehicles, are described in terms of their SE(3) pose and generalized velocity, and satisfy conservation of energy principles. This paper proposes a Hamiltonian formulation over the SE(3) manifold of the structure of a neural ordinary differential equation (ODE) network to approximate the dynamics of a rigid body. In contrast to a black-box ODE network, our formulation guarantees total energy conservation by construction. We develop energy shaping and damping injection control for the learned, potentially under-actuated SE(3) Hamiltonian dynamics to enable a unified approach for stabilization and trajectory tracking with various platforms, including pendulum, rigid-body, and quadrotor systems.