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Many applications, such as optimization, uncertainty quantification andinverse problems, require repeatedly performing simulations oflarge-dimensional physical systems for different choices of parameters. Thiscan be prohibitively expensive.  In order to save computational cost, one can construct surrogate models byexpressing 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 ofHamiltonian systems, it is crucial to preserve the symplectic structureassociated with the system in order to ensure long-term numerical stability.  Up to this point structure-preserving reductions have largely been limited tolinear transformations. We propose a new neural network architecture in thespirit of autoencoders, which are established tools for dimension reduction andfeature extraction in data science, to obtain more general mappings.  In order to train the network, a non-standard gradient descent approach isapplied that leverages the differential-geometric structure emerging from thenetwork design.  The new architecture is shown to significantly outperform existing designs inaccuracy.

Symplectic Autoencoders for Model Reduction of Hamiltonian Systems