  Non-affine parametric dependencies, nonlinearities and advection-dominated
regimes of the model of interest can result in a slow Kolmogorov n-width decay,
which precludes the realization of efficient reduced-order models based on
linear subspace approximations. Among the possible solutions, there are purely
data-driven methods that leverage autoencoders and their variants to learn a
latent representation of the dynamical system, and then evolve it in time with
another architecture. Despite their success in many applications where standard
linear techniques fail, more has to be done to increase the interpretability of
the results, especially outside the training range and not in regimes
characterized by an abundance of data. Not to mention that none of the
knowledge on the physics of the model is exploited during the predictive phase.
In order to overcome these weaknesses, we implement the non-linear manifold
method introduced by Carlberg et al [37] with hyper-reduction achieved through
reduced over-collocation and teacher-student training of a reduced decoder. We
test the methodology on a 2d non-linear conservation law and a 2d shallow water
models, and compare the results obtained with a purely data-driven method for
which the dynamics is evolved in time with a long-short term memory network.
