Symplectic Autoencoders for Model Reduction of Hamiltonian Systems
Benedikt Brantner Michael Kraus
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.