# Time-adaptive H'enonNets for separable Hamiltonian systems

arXiv: https://arxiv.org/abs/2509.20212 · HTML: https://mlsanity.com/paper/2509.20212/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 23 references that are in the map.
- First posted on arXiv 24 Sep 2025.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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 Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [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)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Adaptable Hamiltonian neural networks](https://mlsanity.com/paper/2102.13235/) (2021)
- [Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings](https://mlsanity.com/paper/2308.01084/) (2023)
- [Time-adaptive SympNets for separable Hamiltonian systems](https://mlsanity.com/paper/2509.16026/) (2025)

## Built on by (most cited first, 1 of 1)

- [Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics](https://mlsanity.com/paper/2607.20235/) (2026)

## Abstract (arXiv)

Measurement data is often sampled irregularly, i.e., not on equidistant time grids. This is also true for Hamiltonian systems. However, existing machine learning methods, which learn symplectic integrators, such as SympNets [1] and H'enonNets [2] still require training data generated by fixed step sizes. To learn time-adaptive symplectic integrators, an extension to SympNets called TSympNets is introduced in [3]. The aim of this work is to do a similar extension for H'enonNets. We propose a novel neural network architecture called T-H'enonNets, which is symplectic by design and can handle adaptive time steps. We also extend the T-H'enonNet architecture to non-autonomous Hamiltonian systems. Additionally, we provide universal approximation theorems for both new architectures for separable Hamiltonian systems and discuss why it is difficult to handle non-separable Hamiltonian systems with the proposed methods. To investigate these theoretical approximation capabilities, we perform different numerical experiments.