Symplectic convolutional neural networks
This work addresses the need for structure-preserving neural networks in physics simulations, offering a domain-specific improvement for symplectic systems.
The paper tackled the problem of designing convolutional neural networks (CNNs) that preserve symplectic structure, by proposing a new symplectic CNN architecture using symplectic neural networks and tensor techniques, and demonstrated its performance on wave, nonlinear Schrödinger, and sine-Gordon equations, showing it outperforms linear symplectic autoencoders.
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.