# Learning symplectic model reduction based on an approximation theorem of symplectic embeddings

arXiv: https://arxiv.org/abs/2606.04623 · HTML: https://mlsanity.com/paper/2606.04623/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 61 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Diffusion Autoencoders: Toward a Meaningful and Decodable Representation](https://mlsanity.com/paper/2111.15640/) (2021)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Contrastive Variational Autoencoder Enhances Salient Features](https://mlsanity.com/paper/1902.04601/) (2019)
- [Constructing Custom Thermodynamics Using Deep Learning](https://mlsanity.com/paper/2308.04119/) (2023)
- [Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds](https://mlsanity.com/paper/2305.15490/) (2023)
- [Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-nets](https://mlsanity.com/paper/2303.17824/) (2023)
- [Symplectic Autoencoders for Model Reduction of Hamiltonian Systems](https://mlsanity.com/paper/2312.10004/) (2023)
- [Approximation capabilities of measure-preserving neural networks](https://mlsanity.com/paper/2106.10911/) (2021)
- [Continuity-Preserving Convolutional Autoencoders for Learning Continuous Latent Dynamical Models from Images](https://mlsanity.com/paper/2502.00754/) (2025)
- [Identifiable learning of dissipative dynamics](https://mlsanity.com/paper/2510.24160/) (2025)
- [Data-driven intrinsic localized mode detection and classification in one-dimensional crystal lattice model](https://mlsanity.com/paper/2110.12870/) (2021)
- [Reduced-order modeling of Hamiltonian dynamics based on symplectic neural networks](https://mlsanity.com/paper/2508.11911/) (2025)
- [Symplectic convolutional neural networks](https://mlsanity.com/paper/2508.19842/) (2025)

## Abstract (arXiv)

High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics that evolve on symplectic manifolds. Although deep learning provides powerful tools for constructing its low-dimensional surrogates from data, the intrinsic symplectic structure is easily destroyed during model reduction. As a result, a standard autoencoder may produce latent coordinates that do not support a Hamiltonian flow, leading to unstable long-time prediction. In this paper, we first establish a universal approximation theorem for symplectic embeddings. And based on the theory, we propose symplecticity-preserving autoencoders (SpAE), in which the decoder is parameterized as a symplectic embedding and the encoder is constructed as the corresponding symplectic projection. This architecture is expressive enough to approximate nonlinear symplectic embeddings and the corresponding symplectic projection, preserves the symplectic structure exactly by construction, and can be trained by standard unconstrained optimization, thereby improving both reconstruction and prediction accuracy. Extensive experiments on high-dimensional lattice and particle systems demonstrate the effectiveness of the proposed method.