# Symplectic Learning for Hamiltonian Neural Networks

arXiv: https://arxiv.org/abs/2106.11753 · HTML: https://mlsanity.com/paper/2106.11753/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 41 references that are in the map.
- First posted on arXiv 22 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge](https://mlsanity.com/paper/1711.07970/) (2017)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [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)
- [Sparse Symplectically Integrated Neural Networks](https://mlsanity.com/paper/2006.12972/) (2020)
- [EM-like Learning Chaotic Dynamics from Noisy and Partial Observations](https://mlsanity.com/paper/1903.10335/) (2019)

## Built on by (most cited first, 30 of 35)

- [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)
- [Pseudo-Hamiltonian neural networks for learning partial differential equations](https://mlsanity.com/paper/2304.14374/) (2023)
- [Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2308.15349/) (2023)
- [Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems](https://mlsanity.com/paper/2211.16006/) (2022)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Approximation of nearly-periodic symplectic maps via structure-preserving neural networks](https://mlsanity.com/paper/2210.05087/) (2022)
- [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 Networks Based on Dynamical Systems](https://mlsanity.com/paper/2408.09821/) (2024)
- [Noether's razor: Learning Conserved Quantities](https://mlsanity.com/paper/2410.08087/) (2024)
- [Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraints](https://mlsanity.com/paper/2505.20515/) (2025)
- [Learning Hamiltonian Systems with Mono-Implicit Runge-Kutta Methods](https://mlsanity.com/paper/2303.03769/) (2023)
- [CLPNets: Coupled Lie-Poisson Neural Networks for Multi-Part Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2408.16160/) (2024)
- [Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflation](https://mlsanity.com/paper/2410.05445/) (2024)
- [Data-driven particle dynamics: Structure-preserving coarse-graining for emergent behavior in non-equilibrium systems](https://mlsanity.com/paper/2508.12569/) (2025)
- [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)
- [Neural non-canonical Hamiltonian dynamics for long-time simulations](https://mlsanity.com/paper/2510.01788/) (2025)
- [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)
- [What's Next? Predicting Hamiltonian Dynamics from Discrete Observations of a Vector Field](https://mlsanity.com/paper/2312.08944/) (2023)
- [Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling](https://mlsanity.com/paper/2505.22527/) (2025)
- [Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Network](https://mlsanity.com/paper/2507.14467/) (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)
- [CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems](https://mlsanity.com/paper/2607.03339/) (2026)
- [Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical response](https://mlsanity.com/paper/2404.17584/) (2024)

## Abstract (arXiv)

Machine learning methods are widely used in the natural sciences to model and predict physical systems from observation data. Yet, they are often used as poorly understood "black boxes," disregarding existing mathematical structure and invariants of the problem. Recently, the proposal of Hamiltonian Neural Networks (HNNs) took a first step towards a unified "gray box" approach, using physical insight to improve performance for Hamiltonian systems. In this paper, we explore a significantly improved training method for HNNs, exploiting the symplectic structure of Hamiltonian systems with a different loss function. This frees the loss from an artificial lower bound. We mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn. This allows us to prove and numerically analyze the errors made by HNNs which, in turn, renders them fully explainable. Finally, we present a novel post-training correction to obtain the true Hamiltonian only from discretized observation data, up to an arbitrary order.