  With the rapid advance of Machine Learning techniques and the deep increase
of availability of scientific data, data-driven approaches have started to
become progressively popular across science, causing a fundamental shift in the
scientific method after proving to be powerful tools with a direct impact in
many areas of society. Nevertheless, when attempting to analyze dynamics of
complex multiscale systems, the usage of standard Deep Neural Networks (DNNs)
and even standard Physics-Informed Neural Networks (PINNs) may lead to
incorrect inferences and predictions, due to the presence of small scales
leading to reduced or simplified models in the system that have to be applied
consistently during the learning process. In this Chapter, we will address
these issues in light of recent results obtained in the development of
Asymptotic-Preserving Neural Networks (APNNs) for hyperbolic models with
diffusive scaling. Several numerical tests show how APNNs provide considerably
better results with respect to the different scales of the problem when
compared with standard DNNs and PINNs, especially when analyzing scenarios in
which only little and scattered information is available.
