用辛结构设计卷积网络，提升物理方程模拟精度。
Symplectic convolutional neural networks
- 引入辛卷积层的数学等价形式，保证网络保持辛结构。
- 在波方程、非线性薛定谔方程等任务中优于传统方法。
- 适合需要长期稳定物理模拟的科研与工程场景。
我们提出一种新型辛卷积神经网络（CNN）架构，结合辛神经网络、恰当辛分解和张量技术。首先，给出卷积层的数学等价形式；随后，利用辛神经网络参数化卷积层，确保其保持辛结构。为构建完整自编码器，引入辛池化层。在波方程、非线性薛定谔（NLS）方程和 sine-Gordon 方程三个例子上验证性能。数值结果表明，该辛CNN优于通过恰当辛分解获得的线性辛自编码器。
原文摘要 · Abstract (English)
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.
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