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Deep Learning is having a remarkable impact on the design of Reduced OrderModels (ROMs) for Partial Differential Equations (PDEs), where it is exploitedas a powerful tool for tackling complex problems for which classical methodsmight fail. In this respect, deep autoencoders play a fundamental role, as theyprovide an extremely flexible tool for reducing the dimensionality of a givenproblem by leveraging on the nonlinear capabilities of neural networks. Indeed,starting from this paradigm, several successful approaches have already beendeveloped, which are here referred to as Deep Learning-based ROMs (DL-ROMs).Nevertheless, when it comes to stochastic problems parameterized by randomfields, the current understanding of DL-ROMs is mostly based on empiricalevidence: in fact, their theoretical analysis is currently limited to the caseof PDEs depending on a finite number of (deterministic) parameters. The purposeof this work is to extend the existing literature by providing some theoreticalinsights about the use of DL-ROMs in the presence of stochasticity generated byrandom fields. In particular, we derive explicit error bounds that can guidedomain practitioners when choosing the latent dimension of deep autoencoders.We evaluate the practical usefulness of our theory by means of numericalexperiments, showing how our analysis can significantly impact the performanceof DL-ROMs.

On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields