Symplectic convolutional neural networks

2025-08-27

AI Summary

Conventional CNNs fail to preserve the symplectic geometric structure when solving Hamiltonian partial differential equations—such as the wave equation, nonlinear Schrödinger equation, and sine-Gordon equation—leading to degraded long-term dynamical fidelity. Method: We propose the first Symplectic Convolutional Neural Network (Symplectic-CNN), integrating symplectic neural networks with tensor decomposition to enforce symplectic parameterization of convolutional kernels, and introducing the first differentiable symplectic pooling layer to construct an end-to-end symplectic-preserving autoencoder. Contributions/Results: (1) First embedding of symplectic constraints into convolutional architectures; (2) Design of symplectic pooling to ensure phase-space volume conservation; (3) Adoption of proper symplectic decomposition for numerical stability. Experiments demonstrate that our model significantly outperforms linear symplectic autoencoders across multiple Hamiltonian PDEs, achieving both high accuracy and superior long-term dynamical preservation.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Non-convex Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Assisted, interactive, and conversational search
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Designing symplectic convolutional neural networks for physics
• Ensuring convolution layers remain symplectic through parameterization
• Testing performance on wave and nonlinear equations

Innovation

• Symplectic CNN architecture using neural networks
• Parameterizing layers to maintain symplectic properties
• Introducing symplectic pooling for complete autoencoder