# Symplectic convolutional neural networks

arXiv: https://arxiv.org/abs/2508.19842 · HTML: https://mlsanity.com/paper/2508.19842/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 39 references that are in the map.
- First posted on arXiv 27 Aug 2025.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep Speech: Scaling up end-to-end speech recognition](https://mlsanity.com/paper/1412.5567/) (2014)
- [Towards Understanding the Invertibility of Convolutional Neural Networks](https://mlsanity.com/paper/1705.08664/) (2017)
- [Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings](https://mlsanity.com/paper/2308.01084/) (2023)
- [Symplectic Autoencoders for Model Reduction of Hamiltonian Systems](https://mlsanity.com/paper/2312.10004/) (2023)
- [Deep Learning for Structure-Preserving Universal Stable Koopman-Inspired Embeddings for Nonlinear Canonical Hamiltonian Dynamics](https://mlsanity.com/paper/2308.13835/) (2023)
- [Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy](https://mlsanity.com/paper/2503.02273/) (2025)

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

- [Learning symplectic model reduction based on an approximation theorem of symplectic embeddings](https://mlsanity.com/paper/2606.04623/) (2026)

## Abstract (arXiv)

We propose a new symplectic convolutional neural network (CNN) architecture by leveraging symplectic neural networks, proper symplectic decomposition, and tensor techniques. Specifically, we first introduce a mathematically equivalent form of the convolution layer and then, using symplectic neural networks, we demonstrate a way to parameterize the layers of the CNN to ensure that the convolution layer remains symplectic. To construct a complete autoencoder, we introduce a symplectic pooling layer. We demonstrate the performance of the proposed neural network on three examples: the wave equation, the nonlinear Schr"odinger (NLS) equation, and the sine-Gordon equation. The numerical results indicate that the symplectic CNN outperforms the linear symplectic autoencoder obtained via proper symplectic decomposition.