# Nonseparable Symplectic Neural Networks

arXiv: https://arxiv.org/abs/2010.12636 · HTML: https://mlsanity.com/paper/2010.12636/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2020.

## Built on by (most cited first, 30 of 31)

- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Deconstructing the Inductive Biases of Hamiltonian Neural Networks](https://mlsanity.com/paper/2202.04836/) (2022)
- [Which priors matter? Benchmarking models for learning latent dynamics](https://mlsanity.com/paper/2111.05458/) (2021)
- [Locally-symplectic neural networks for learning volume-preserving dynamics](https://mlsanity.com/paper/2109.09151/) (2021)
- [SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision](https://mlsanity.com/paper/2111.05986/) (2021)
- [Physics-Informed Learning Using Hamiltonian Neural Networks with Output Error Noise Models](https://mlsanity.com/paper/2305.01338/) (2023)
- [Learning Trajectories of Hamiltonian Systems with Neural Networks](https://mlsanity.com/paper/2204.05077/) (2022)
- [Symplectic Neural Networks Based on Dynamical Systems](https://mlsanity.com/paper/2408.09821/) (2024)
- [Symplectic Neural Flows for Modeling and Discovery](https://mlsanity.com/paper/2412.16787/) (2024)
- [Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models](https://mlsanity.com/paper/2209.07646/) (2022)
- [CLPNets: Coupled Lie-Poisson Neural Networks for Multi-Part Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2408.16160/) (2024)
- [KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss](https://mlsanity.com/paper/2102.11923/) (2021)
- [VPNets: Volume-preserving neural networks for learning source-free dynamics](https://mlsanity.com/paper/2204.13843/) (2022)
- [Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling](https://mlsanity.com/paper/2401.12476/) (2024)
- [Learning Generalized Hamiltonians using fully Symplectic Mappings](https://mlsanity.com/paper/2409.11138/) (2024)
- [Training Hamiltonian neural networks without backpropagation](https://mlsanity.com/paper/2411.17511/) (2024)
- [Rapid training of Hamiltonian graph networks using random features](https://mlsanity.com/paper/2506.06558/) (2025)
- [Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach](https://mlsanity.com/paper/2509.24627/) (2025)
- [Symplectically Integrated Symbolic Regression of Hamiltonian Dynamical Systems](https://mlsanity.com/paper/2209.01521/) (2022)
- [Data-driven discovery of non-Newtonian astronomy via learning non-Euclidean Hamiltonian](https://mlsanity.com/paper/2210.00090/) (2022)
- [Learning Physical Systems: Symplectification via Gauge Fixing in Dirac Structures](https://mlsanity.com/paper/2506.18812/) (2025)
- [Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Network](https://mlsanity.com/paper/2507.14467/) (2025)
- [Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics](https://mlsanity.com/paper/2509.13425/) (2025)
- [Time-adaptive SympNets for separable Hamiltonian systems](https://mlsanity.com/paper/2509.16026/) (2025)
- [Time-adaptive H'enonNets for separable Hamiltonian systems](https://mlsanity.com/paper/2509.20212/) (2025)
- [Controlled oscillation modeling using port-Hamiltonian neural networks](https://mlsanity.com/paper/2602.15704/) (2026)
- [Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data](https://mlsanity.com/paper/2604.17470/) (2026)
- [Data-Driven Computing Methods for Nonlinear Physics Systems with Geometric Constraints](https://mlsanity.com/paper/2406.16956/) (2024)
- [Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data](https://mlsanity.com/paper/2604.23606/) (2026)

## Abstract (arXiv)

Predicting the behaviors of Hamiltonian systems has been drawing increasing attention in scientific machine learning. However, the vast majority of the literature was focused on predicting separable Hamiltonian systems with their kinematic and potential energy terms being explicitly decoupled while building data-driven paradigms to predict nonseparable Hamiltonian systems that are ubiquitous in fluid dynamics and quantum mechanics were rarely explored. The main computational challenge lies in the effective embedding of symplectic priors to describe the inherently coupled evolution of position and momentum, which typically exhibits intricate dynamics. To solve the problem, we propose a novel neural network architecture, Nonseparable Symplectic Neural Networks (NSSNNs), to uncover and embed the symplectic structure of a nonseparable Hamiltonian system from limited observation data. The enabling mechanics of our approach is an augmented symplectic time integrator to decouple the position and momentum energy terms and facilitate their evolution. We demonstrated the efficacy and versatility of our method by predicting a wide range of Hamiltonian systems, both separable and nonseparable, including chaotic vortical flows. We showed the unique computational merits of our approach to yield long-term, accurate, and robust predictions for large-scale Hamiltonian systems by rigorously enforcing symplectomorphism.