On the latent dimension of deep autoencoders for reduced order modeling
  of PDEs parametrized by random fields
Nicola Rares Franco Daniel Fraulin Andrea Manzoni Paolo Zunino
Deep Learning is having a remarkable impact on the design of Reduced Order
Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited
as a powerful tool for tackling complex problems for which classical methods
might fail. In this respect, deep autoencoders play a fundamental role, as they
provide an extremely flexible tool for reducing the dimensionality of a given
problem by leveraging on the nonlinear capabilities of neural networks. Indeed,
starting from this paradigm, several successful approaches have already been
developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs).
Nevertheless, when it comes to stochastic problems parameterized by random
fields, the current understanding of DL-ROMs is mostly based on empirical
evidence: in fact, their theoretical analysis is currently limited to the case
of PDEs depending on a finite number of (deterministic) parameters. The purpose
of this work is to extend the existing literature by providing some theoretical
insights about the use of DL-ROMs in the presence of stochasticity generated by
random fields. In particular, we derive explicit error bounds that can guide
domain practitioners when choosing the latent dimension of deep autoencoders.
We evaluate the practical usefulness of our theory by means of numerical
experiments, showing how our analysis can significantly impact the performance
of DL-ROMs.