Symplectic convolutional neural networks

TL;DR

This paper introduces a novel symplectic CNN architecture that preserves symplectic structure through specialized layers, demonstrating superior performance on wave-related equations compared to traditional methods.

Contribution

The paper develops a symplectic CNN architecture with symplectic layers and pooling, ensuring structure preservation and improved modeling of wave equations.

Findings

Outperforms linear symplectic autoencoders on wave equations

Ensures symplectic structure in CNN layers

Demonstrates effectiveness on wave, NLS, and sine-Gordon equations

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 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.