# Symplectic Autoencoders for Model Reduction of Hamiltonian Systems

arXiv: https://arxiv.org/abs/2312.10004 · HTML: https://mlsanity.com/paper/2312.10004/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 35 references that are in the map.
- First posted on arXiv 15 Dec 2023.

## Builds on (most cited first, 6 of 6)

- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [A Differentiable Programming System to Bridge Machine Learning and Scientific Computing](https://mlsanity.com/paper/1907.07587/) (2019)
- [Momentum Stiefel Optimizer, with Applications to Suitably-Orthogonal Attention, and Optimal Transport](https://mlsanity.com/paper/2205.14173/) (2022)
- [Riemannian Optimization on the Symplectic Stiefel Manifold](https://mlsanity.com/paper/2006.15226/) (2020)
- [Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications](https://mlsanity.com/paper/2211.09481/) (2022)
- [Generalizing Adam to Manifolds for Efficiently Training Transformers](https://mlsanity.com/paper/2305.16901/) (2023)

## Built on by (most cited first, 5 of 5)

- [Volume-Preserving Transformers for Learning Time Series Data with Structure](https://mlsanity.com/paper/2312.11166/) (2023)
- [Data-driven identification of latent port-Hamiltonian systems](https://mlsanity.com/paper/2408.08185/) (2024)
- [Symplectic convolutional neural networks](https://mlsanity.com/paper/2508.19842/) (2025)
- [Learning symplectic model reduction based on an approximation theorem of symplectic embeddings](https://mlsanity.com/paper/2606.04623/) (2026)
- [Learning Deterministic and Stochastic Forced Hamiltonian Systems](https://mlsanity.com/paper/2608.19688/) (2026)

## Abstract (arXiv)

Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively expensive. In order to save computational cost, one can construct surrogate models by expressing the system in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. Up to this point structure-preserving reductions have largely been limited to linear transformations. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction and feature extraction in data science, to obtain more general mappings. In order to train the network, a non-standard gradient descent approach is applied that leverages the differential-geometric structure emerging from the network design. The new architecture is shown to significantly outperform existing designs in accuracy.