# Sparse Symplectically Integrated Neural Networks

arXiv: https://arxiv.org/abs/2006.12972 · HTML: https://mlsanity.com/paper/2006.12972/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2020.

## Built on by (most cited first, 20 of 20)

- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces](https://mlsanity.com/paper/2206.02660/) (2022)
- [Learning Hamiltonians of constrained mechanical systems](https://mlsanity.com/paper/2201.13254/) (2022)
- [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022)
- [On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solver](https://mlsanity.com/paper/2208.03680/) (2022)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision](https://mlsanity.com/paper/2111.05986/) (2021)
- [Learning Trajectories of Hamiltonian Systems with Neural Networks](https://mlsanity.com/paper/2204.05077/) (2022)
- [Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators](https://mlsanity.com/paper/2306.03548/) (2023)
- [Symplectic Neural Flows for Modeling and Discovery](https://mlsanity.com/paper/2412.16787/) (2024)
- [Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflation](https://mlsanity.com/paper/2410.05445/) (2024)
- [KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss](https://mlsanity.com/paper/2102.11923/) (2021)
- [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)
- [Pseudo-Hamiltonian system identification](https://mlsanity.com/paper/2305.06920/) (2023)
- [H-FEX: A Symbolic Learning Method for Hamiltonian Systems](https://mlsanity.com/paper/2506.20607/) (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)

## Abstract (arXiv)

We introduce Sparse Symplectically Integrated Neural Networks (SSINNs), a novel model for learning Hamiltonian dynamical systems from data. SSINNs combine fourth-order symplectic integration with a learned parameterization of the Hamiltonian obtained using sparse regression through a mathematically elegant function space. This allows for interpretable models that incorporate symplectic inductive biases and have low memory requirements. We evaluate SSINNs on four classical Hamiltonian dynamical problems: the H'enon-Heiles system, nonlinearly coupled oscillators, a multi-particle mass-spring system, and a pendulum system. Our results demonstrate promise in both system prediction and conservation of energy, often outperforming the current state-of-the-art black-box prediction techniques by an order of magnitude. Further, SSINNs successfully converge to true governing equations from highly limited and noisy data, demonstrating potential applicability in the discovery of new physical governing equations.