# Learning Deterministic and Stochastic Forced Hamiltonian Systems

arXiv: https://arxiv.org/abs/2608.19688 · HTML: https://mlsanity.com/paper/2608.19688/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 91 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates](https://mlsanity.com/paper/2008.10546/) (2020)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [Modeling Unknown Stochastic Dynamical System via Autoencoder](https://mlsanity.com/paper/2312.10001/) (2023)
- [Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme](https://mlsanity.com/paper/2105.08449/) (2021)
- [Generalized Lagrangian Neural Networks](https://mlsanity.com/paper/2401.03728/) (2024)
- [DynGMA: a robust approach for learning stochastic differential equations from data](https://mlsanity.com/paper/2402.14475/) (2024)
- [Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems](https://mlsanity.com/paper/2211.16006/) (2022)
- [Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics](https://mlsanity.com/paper/2505.20370/) (2025)
- [Forced Variational Integrator Networks for Prediction and Control of Mechanical Systems](https://mlsanity.com/paper/2106.02973/) (2021)
- [Symplectic Autoencoders for Model Reduction of Hamiltonian Systems](https://mlsanity.com/paper/2312.10004/) (2023)
- [Neural non-canonical Hamiltonian dynamics for long-time simulations](https://mlsanity.com/paper/2510.01788/) (2025)
- [Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Network](https://mlsanity.com/paper/2507.14467/) (2025)

## Abstract (arXiv)

We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a Cr convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.