The reduction of Hamiltonian systems aims to build smaller reduced models, valid over a certain range of time and parameters, in order to reduce computing time. By maintaining the Hamiltonian structure in the reduced model, certain long-term stability properties can be preserved. In this paper, we propose a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks. Their training is coupled in order to simultaneously learn the encoder-decoder operators and the reduced dynamics. Several test cases on non-linear wave dynamics show that the method has better reduction properties than standard linear Hamiltonian reduction methods.

翻译：哈密顿系统约化的目标是在特定时间和参数范围内构建更小、有效的降阶模型，从而降低计算成本。通过在降阶模型中保持哈密顿结构，可以保留某些长期稳定性特征。本文提出一种针对偏微分方程空间离散化模型建立的非线性约化方法：该方法基于卷积自编码器与哈密顿神经网络。通过耦合训练这两类网络，可同步学习编码器-解码器算子与降阶动力学。多个非线性波动动力学算例表明，该方法相较于标准线性哈密顿约化方法具有更优的约化性能。