# Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling

arXiv: https://arxiv.org/abs/2505.22527 · HTML: https://mlsanity.com/paper/2505.22527/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 30 references that are in the map.
- First posted on arXiv 28 May 2025.

## Builds on (most cited first, 12 of 12)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Locally-symplectic neural networks for learning volume-preserving dynamics](https://mlsanity.com/paper/2109.09151/) (2021)

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

- [FlowPET: Physics-Informed Symplectic Flow Matching for Low-Count PET Reconstruction](https://mlsanity.com/paper/2607.11104/) (2026)

## Abstract (arXiv)

We introduce the Symplectic Generative Network (SGN), a deep generative model that leverages Hamiltonian mechanics to construct an invertible, volume-preserving mapping between a latent space and the data space. By endowing the latent space with a symplectic structure and modeling data generation as the time evolution of a Hamiltonian system, SGN achieves exact likelihood evaluation without incurring the computational overhead of Jacobian determinant calculations. In this work, we provide a rigorous mathematical foundation for SGNs through a comprehensive theoretical framework that includes: (i) complete proofs of invertibility and volume preservation, (ii) a formal complexity analysis with theoretical comparisons to Variational Autoencoders and Normalizing Flows, (iii) strengthened universal approximation results with quantitative error bounds, (iv) an information-theoretic analysis based on the geometry of statistical manifolds, and (v) an extensive stability analysis with adaptive integration guarantees. These contributions highlight the fundamental advantages of SGNs and establish a solid foundation for future empirical investigations and applications to complex, high-dimensional data.