Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively expensive. In order to save computational cost, one can construct surrogate models by expressing the system in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. Up to this point structure-preserving reductions have largely been limited to linear transformations. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction and feature extraction in data science, to obtain more general mappings. In order to train the network, a non-standard gradient descent approach is applied that leverages the differential-geometric structure emerging from the network design. The new architecture is shown to significantly outperform existing designs in accuracy.

翻译：许多应用，如优化、不确定性量化和反问题，需要针对不同参数选择重复执行高维物理系统的模拟。这可能代价高昂。为了节省计算成本，可以通过从训练数据中获得的低维基底来表示系统，从而构建代理模型。这称为模型降阶。过去的研究表明，在对哈密顿系统进行模型降阶时，保留与系统相关的辛结构对于确保长期数值稳定性至关重要。迄今为止，保结构降阶主要限于线性变换。我们提出了一种新的神经网络架构，其思想源于自编码器——数据科学中用于降维和特征提取的成熟工具，以获得更通用的映射。为了训练该网络，采用了一种非标准梯度下降方法，该方法利用了网络设计中涌现的微分几何结构。实验表明，新架构在精度上显著优于现有设计。