# Neural non-canonical Hamiltonian dynamics for long-time simulations

arXiv: https://arxiv.org/abs/2510.01788 · HTML: https://mlsanity.com/paper/2510.01788/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 50 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 27 of 27)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws](https://mlsanity.com/paper/2303.03192/) (2023)
- [The Elements of Differentiable Programming](https://mlsanity.com/paper/2403.14606/) (2024)
- [Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks](https://mlsanity.com/paper/2012.03133/) (2020)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Hypersolvers: Toward Fast Continuous-Depth Models](https://mlsanity.com/paper/2007.09601/) (2020)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces](https://mlsanity.com/paper/2206.02660/) (2022)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [Pseudo-Hamiltonian neural networks for learning partial differential equations](https://mlsanity.com/paper/2304.14374/) (2023)
- [DynGMA: a robust approach for learning stochastic differential equations from data](https://mlsanity.com/paper/2402.14475/) (2024)
- [Approximation of nearly-periodic symplectic maps via structure-preserving neural networks](https://mlsanity.com/paper/2210.05087/) (2022)
- [Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery](https://mlsanity.com/paper/2211.10830/) (2022)
- [ModLaNets: Learning Generalisable Dynamics via Modularity and Physical Inductive Bias](https://mlsanity.com/paper/2206.12325/) (2022)
- [Latent Dynamics Networks (LDNets): learning the intrinsic dynamics of spatio-temporal processes](https://mlsanity.com/paper/2305.00094/) (2023)
- [Symplectic Neural Networks Based on Dynamical Systems](https://mlsanity.com/paper/2408.09821/) (2024)

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

- [Learning Deterministic and Stochastic Forced Hamiltonian Systems](https://mlsanity.com/paper/2608.19688/) (2026)
- [The limits of exactness: On the failure of automatic differentiation in physics-informed machine learning](https://mlsanity.com/paper/2609.33078/) (2026)

## Abstract (arXiv)

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.