  When investigating epidemic dynamics through differential models, the
parameters needed to understand the phenomenon and to simulate forecast
scenarios require a delicate calibration phase, often made even more
challenging by the scarcity and uncertainty of the observed data reported by
official sources. In this context, Physics-Informed Neural Networks (PINNs), by
embedding the knowledge of the differential model that governs the physical
phenomenon in the learning process, can effectively address the inverse and
forward problem of data-driven learning and solving the corresponding epidemic
problem. In many circumstances, however, the spatial propagation of an
infectious disease is characterized by movements of individuals at different
scales governed by multiscale PDEs. This reflects the heterogeneity of a region
or territory in relation to the dynamics within cities and in neighboring
zones. In presence of multiple scales, a direct application of PINNs generally
leads to poor results due to the multiscale nature of the differential model in
the loss function of the neural network. To allow the neural network to operate
uniformly with respect to the small scales, it is desirable that the neural
network satisfies an Asymptotic-Preservation (AP) property in the learning
process. To this end, we consider a new class of AP Neural Networks (APNNs) for
multiscale hyperbolic transport models of epidemic spread that, thanks to an
appropriate AP formulation of the loss function, is capable to work uniformly
at the different scales of the system. A series of numerical tests for
different epidemic scenarios confirms the validity of the proposed approach,
highlighting the importance of the AP property in the neural network when
dealing with multiscale problems especially in presence of sparse and partially
observed systems.
