# Locally-symplectic neural networks for learning volume-preserving dynamics

arXiv: https://arxiv.org/abs/2109.09151 · HTML: https://mlsanity.com/paper/2109.09151/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 36 references that are in the map.
- First posted on arXiv 19 Sep 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [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)
- [Structure preserving deep learning](https://mlsanity.com/paper/2006.03364/) (2020)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Learning Physical Constraints with Neural Projections](https://mlsanity.com/paper/2006.12745/) (2020)
- [Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Design](https://mlsanity.com/paper/2105.13205/) (2021)
- [Learning Symmetries of Classical Integrable Systems](https://mlsanity.com/paper/1906.04645/) (2019)
- [Solving Newton's Equations of Motion with Large Timesteps using Recurrent Neural Networks based Operators](https://mlsanity.com/paper/2004.06493/) (2020)
- [Approximation capabilities of measure-preserving neural networks](https://mlsanity.com/paper/2106.10911/) (2021)

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

- [Dynamical systems' based neural networks](https://mlsanity.com/paper/2210.02373/) (2022)
- [Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2308.15349/) (2023)
- [Symplectic Neural Networks Based on Dynamical Systems](https://mlsanity.com/paper/2408.09821/) (2024)
- [Volume-Preserving Transformers for Learning Time Series Data with Structure](https://mlsanity.com/paper/2312.11166/) (2023)
- [VPNets: Volume-preserving neural networks for learning source-free dynamics](https://mlsanity.com/paper/2204.13843/) (2022)
- [Predictions Based on Pixel Data: Insights from PDEs and Finite Differences](https://mlsanity.com/paper/2305.00723/) (2023)
- [Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling](https://mlsanity.com/paper/2505.22527/) (2025)
- [Time-adaptive SympNets for separable Hamiltonian systems](https://mlsanity.com/paper/2509.16026/) (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)

We propose locally-symplectic neural networks LocSympNets for learning the flow of phase volume-preserving dynamics. The construction of LocSympNets stems from the theorem of the local Hamiltonian description of the divergence-free vector field and the splitting methods based on symplectic integrators. Symplectic gradient modules of the recently proposed symplecticity-preserving neural networks SympNets are used to construct invertible locally-symplectic modules. To further preserve properties of the flow of a dynamical system LocSympNets are extended to symmetric locally-symplectic neural networks SymLocSympNets, such that the inverse of SymLocSympNets is equal to the feed-forward propagation of SymLocSympNets with the negative time step, which is a general property of the flow of a dynamical system. LocSympNets and SymLocSympNets are studied numerically considering learning linear and nonlinear volume-preserving dynamics. We demonstrate learning of linear traveling wave solutions to the semi-discretized advection equation, periodic trajectories of the Euler equations of the motion of a free rigid body, and quasi-periodic solutions of the charged particle motion in an electromagnetic field. LocSympNets and SymLocSympNets can learn linear and nonlinear dynamics to a high degree of accuracy even when random noise is added to the training data. When learning a single trajectory of the rigid body dynamics locally-symplectic neural networks can learn both quadratic invariants of the system with absolute relative errors below 1%. In addition, SymLocSympNets produce qualitatively good long-time predictions, when the learning of the whole system from randomly sampled data is considered. LocSympNets and SymLocSympNets can produce accurate short-time predictions of quasi-periodic solutions, which is illustrated in the example of the charged particle motion in an electromagnetic field.