# Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries

arXiv: https://arxiv.org/abs/2308.15349 · HTML: https://mlsanity.com/paper/2308.15349/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 61 references that are in the map.
- First posted on arXiv 29 Aug 2023.

## Builds on (most cited first, 13 of 13)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [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)
- [Predicting the Computational Cost of Deep Learning Models](https://mlsanity.com/paper/1811.11880/) (2018)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [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)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Adaptable Hamiltonian neural networks](https://mlsanity.com/paper/2102.13235/) (2021)
- [Locally-symplectic neural networks for learning volume-preserving dynamics](https://mlsanity.com/paper/2109.09151/) (2021)

## Built on by (most cited first, 9 of 9)

- [Symmetry Preservation in Hamiltonian Systems: Simulation and Learning](https://mlsanity.com/paper/2308.16331/) (2023)
- [Meta-learning Structure-Preserving Dynamics](https://mlsanity.com/paper/2508.11205/) (2025)
- [NN-OpInf: an operator inference approach using structure-preserving composable neural networks](https://mlsanity.com/paper/2603.08488/) (2026)
- [Nonlinear GENERIC-Embedded Neural Networks (N-GENNs): Learning GENERIC dynamics with non-quadratic dissipation potentials](https://mlsanity.com/paper/2605.09058/) (2026)
- [Learning Nonholonomic Dynamics with Constraint Discovery](https://mlsanity.com/paper/2410.15201/) (2024)
- [Variational Neural Networks for Observable Thermodynamics (V-NOTS)](https://mlsanity.com/paper/2509.09899/) (2025)
- [Can Transformers overcome the lack of data in the simulation of history-dependent flows?](https://mlsanity.com/paper/2512.16305/) (2025)
- [Symplectic Representation of Legendre Dynamics](https://mlsanity.com/paper/2512.19409/) (2025)
- [Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics](https://mlsanity.com/paper/2607.28939/) (2026)

## Abstract (arXiv)

An accurate data-based prediction of the long-term evolution of Hamiltonian systems requires a network that preserves the appropriate structure under each time step. Every Hamiltonian system contains two essential ingredients: the Poisson bracket and the Hamiltonian. Hamiltonian systems with symmetries, whose paradigm examples are the Lie-Poisson systems, have been shown to describe a broad category of physical phenomena, from satellite motion to underwater vehicles, fluids, geophysical applications, complex fluids, and plasma physics. The Poisson bracket in these systems comes from the symmetries, while the Hamiltonian comes from the underlying physics. We view the symmetry of the system as primary, hence the Lie-Poisson bracket is known exactly, whereas the Hamiltonian is regarded as coming from physics and is considered not known, or known approximately. Using this approach, we develop a network based on transformations that exactly preserve the Poisson bracket and the special functions of the Lie-Poisson systems (Casimirs) to machine precision. We present two flavors of such systems: one, where the parameters of transformations are computed from data using a dense neural network (LPNets), and another, where the composition of transformations is used as building blocks (G-LPNets). We also show how to adapt these methods to a larger class of Poisson brackets. We apply the resulting methods to several examples, such as rigid body (satellite) motion, underwater vehicles, a particle in a magnetic field, and others. The methods developed in this paper are important for the construction of accurate data-based methods for simulating the long-term dynamics of physical systems.