# Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics

arXiv: https://arxiv.org/abs/2607.28939 · HTML: https://mlsanity.com/paper/2607.28939/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 62 references that are in the map.
- First posted on arXiv 31 Jul 2026.

## Builds on (most cited first, 24 of 24)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [Guaranteed Discovery of Control-Endogenous Latent States with Multi-Step Inverse Models](https://mlsanity.com/paper/2207.08229/) (2022)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks](https://mlsanity.com/paper/2012.03133/) (2020)
- [Real-time optimal control via Deep Neural Networks: study on landing problems](https://mlsanity.com/paper/1610.08668/) (2016)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [Generative ODE Modeling with Known Unknowns](https://mlsanity.com/paper/2003.10775/) (2020)
- [Object Segmentation-Assisted Inter Prediction for Versatile Video Coding](https://mlsanity.com/paper/2403.11694/) (2024)
- [Adaptable Hamiltonian neural networks](https://mlsanity.com/paper/2102.13235/) (2021)
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- [Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems](https://mlsanity.com/paper/2404.05040/) (2024)
- [Neural Video Compression with Context Modulation](https://mlsanity.com/paper/2505.14541/) (2025)
- [Locally-symplectic neural networks for learning volume-preserving dynamics](https://mlsanity.com/paper/2109.09151/) (2021)
- [Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2308.15349/) (2023)
- [In-Loop Filtering via Trained Look-Up Tables](https://mlsanity.com/paper/2407.10926/) (2024)
- [Learning to Predict 3D Rotational Dynamics from Images of a Rigid Body with Unknown Mass Distribution](https://mlsanity.com/paper/2308.14666/) (2023)
- [Symmetry Preservation in Hamiltonian Systems: Simulation and Learning](https://mlsanity.com/paper/2308.16331/) (2023)
- [Learning Control-Oriented Dynamical Structure from Data](https://mlsanity.com/paper/2302.02529/) (2023)
- [Optimal Control Problems with Symmetry Breaking Cost Functions](https://mlsanity.com/paper/1701.06973/) (2017)

## Abstract (arXiv)

Structure-preserving neural networks are essential for the long-term prediction of Hamiltonian systems from data. Many important Hamiltonian systems in mechanics and control admit symmetry reduction to Lie--Poisson systems, including rigid bodies, underwater vehicles, fluids, plasmas, and optimal control problems. A fundamental challenge in learning such systems is that their dynamics evolve in momentum variables that are typically unobservable, while available data consist only of observable quantities such as configurations and velocities. In optimal control applications, the situation is further complicated because the latent variables contain unobservable co-states and the Hamiltonian may be degenerate, preventing the existence of a corresponding Lagrangian and rendering the encoder-decoder approaches inapplicable. We introduce Latent Lie--Poisson Neural Networks (LLPNNs), a structure-preserving framework for learning Lie--Poisson dynamics directly from observable data. The proposed approach exploits three geometric ingredients: (i) learning either a Hamiltonian decoder or a pseudo-Lagrangian encoder on the active variables, (ii) constructing latent trajectories through a universal Noether invariant arising from Lie--Poisson symmetry reduction, and (iii) reconstructing observable and latent dynamics through Lie--Poisson flows combined with Magnus-based Lie-group updates. The resulting method preserves the geometric structure and is applicable to both regular and degenerate Hamiltonian systems. We demonstrate the method on three examples: a generalized rigid body on SO(3), Kirchhoff's underwater vehicle on SE(3), and an optimal-control problem for interacting vehicles on SE(2)N. Numerical experiments show excellent long-term predictive accuracy, strong robustness to noise, and competitive performance using only modest datasets and lightweight neural-network architectures.