Symplectic convolutional neural networks
Süleyman Yıldız, Konrad Janik, Peter Benner
TL;DR
This paper introduces a nonlinear symplectic convolutional autoencoder (SympCAE) that preserves Hamiltonian structure in its latent representation by uniting symplectic neural networks (SympNets), proper symplectic decomposition (PSD), and tensor-based convolutional formulations. It develops a mathematically equivalent convolutional framework using Toeplitz and block-Toeplitz representations, then enforces symplecticity through specialized convolutional modules, PSD-like layers, and a symplectic pooling mechanism, culminating in an encoder–decoder architecture that maintains a symplectic flow. The authors demonstrate that SympCAE surpasses PSD-based linear autoencoders on 1D wave and NLS equations and a 2D sine-Gordon equation, achieving markedly lower reconstruction errors at small latent dimensions and enabling accurate long-time predictions via a downstream SympNet. The work highlights a general approach to structure-preserving dimensionality reduction for Hamiltonian and volume-preserving dynamics, with potential extensions to higher dimensions and noisy data.
Abstract
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 Schrödinger (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.