  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.
