# Learning Generalized Hamiltonians using fully Symplectic Mappings

arXiv: https://arxiv.org/abs/2409.11138 · HTML: https://mlsanity.com/paper/2409.11138/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 38 references that are in the map.
- First posted on arXiv 17 Sep 2024.

## Builds on (most cited first, 11 of 11)

- [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)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Learning reversible symplectic dynamics](https://mlsanity.com/paper/2204.12323/) (2022)
- [Symplectic Methods in Deep Learning](https://mlsanity.com/paper/2406.04104/) (2024)
- [Separable Hamiltonian Neural Networks](https://mlsanity.com/paper/2309.01069/) (2023)

## Built on by (most cited first, 2 of 2)

- [GeoHNNs: Geometric Hamiltonian Neural Networks](https://mlsanity.com/paper/2507.15678/) (2025)
- [Controlled oscillation modeling using port-Hamiltonian neural networks](https://mlsanity.com/paper/2602.15704/) (2026)

## Abstract (arXiv)

Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and in particular Hamiltonian Neural Networks have emerged as a mechanism to incorporate structural inductive bias into the NN model. By ensuring physical invariances are conserved, the models exhibit significantly better sample complexity and out-of-distribution accuracy than standard NNs. Learning the Hamiltonian as a function of its canonical variables, typically position and velocity, from sample observations of the system thus becomes a critical task in system identification and long-term prediction of system behavior. However, to truly preserve the long-run physical conservation properties of Hamiltonian systems, one must use symplectic integrators for a forward pass of the system's simulation. While symplectic schemes have been used in the literature, they are thus far limited to situations when they reduce to explicit algorithms, which include the case of separable Hamiltonians or augmented non-separable Hamiltonians. We extend it to generalized non-separable Hamiltonians, and noting the self-adjoint property of symplectic integrators, we bypass computationally intensive backpropagation through an ODE solver. We show that the method is robust to noise and provides a good approximation of the system Hamiltonian when the state variables are sampled from a noisy observation. In the numerical results, we show the performance of the method concerning Hamiltonian reconstruction and conservation, indicating its particular advantage for non-separable systems.