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.

为了解决大维物理系统不同参数选择下的计算成本过高问题，该研究提出了模型简化和神经网络结构的新方法，其中关键是保存系统的辛结构和利用网络设计中的微分几何结构进行训练，该方法在准确性方面表现显著优于现有设计。