# Discovering Phase Space Structure in Learned Hamiltonian Systems

arXiv: https://arxiv.org/abs/2509.23003 · HTML: https://mlsanity.com/paper/2509.23003/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 61 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 30 of 34)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Towards Accurate Generative Models of Video: A New Metric & Challenges](https://mlsanity.com/paper/1812.01717/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [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)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control](https://mlsanity.com/paper/2007.01926/) (2020)
- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [Hamiltonian neural networks for solving equations of motion](https://mlsanity.com/paper/2001.11107/) (2020)
- [LagNetViP: A Lagrangian Neural Network for Video Prediction](https://mlsanity.com/paper/2010.12932/) (2020)
- [Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty](https://mlsanity.com/paper/2003.02359/) (2020)
- [Discovering Conservation Laws using Optimal Transport and Manifold Learning](https://mlsanity.com/paper/2208.14995/) (2022)
- [FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities](https://mlsanity.com/paper/2210.00272/) (2022)
- [Latent Neural Differential Equations for Video Generation](https://mlsanity.com/paper/2011.03864/) (2020)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [Constants of motion network](https://mlsanity.com/paper/2208.10387/) (2022)
- [GAGrasp: Geometric Algebra Diffusion for Dexterous Grasping](https://mlsanity.com/paper/2503.04123/) (2025)
- [Learning to Predict 3D Rotational Dynamics from Images of a Rigid Body with Unknown Mass Distribution](https://mlsanity.com/paper/2308.14666/) (2023)
- [Hamiltonian latent operators for content and motion disentanglement in image sequences](https://mlsanity.com/paper/2112.01641/) (2021)
- [Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery](https://mlsanity.com/paper/2211.10830/) (2022)
- [Learning Color Equivariant Representations](https://mlsanity.com/paper/2406.09588/) (2024)
- [Behavior-Inspired Neural Networks for Relational Inference](https://mlsanity.com/paper/2406.14746/) (2024)
- [Learning Interpretable Dynamics from Images of a Freely Rotating 3D Rigid Body](https://mlsanity.com/paper/2209.11355/) (2022)

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

- [Frequency-Separable Hamiltonian Neural Network for Multi-Timescale Dynamics](https://mlsanity.com/paper/2603.06354/) (2026)
- [Generating Physically Plausible Parachute Dynamics with Deep Generative Modeling](https://mlsanity.com/paper/2607.12143/) (2026)

## Abstract (arXiv)

Mechanics-inspired neural dynamical models improve interpretability and physical consistency, but they usually assume a known configuration space, fixed physical parameters, and deterministic prediction from a prescribed initial condition. We introduce Structured Phase Space GAN (SPS-GAN), a conditional generative model that learns structured dynamics from both Cartesian trajectories and video. SPS-GAN combines a port-Hamiltonian backbone with adversarial conditional generation, allowing conservative, dissipative, and forced systems to be represented in a single architecture. To infer latent mechanical structure from raw observations, we use a cyclic-coordinate loss that recovers both the system degrees of freedom and canonical coordinates without access to the true phase space. Experiments on simulated and real-world systems show that SPS-GAN improves trajectory accuracy over supervised dynamics baselines and video consistency over generative baselines, while consistently recovering the correct latent mechanical coordinates across observation modalities and dynamical regimes.