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When investigating epidemic dynamics through differential models, theparameters needed to understand the phenomenon and to simulate forecastscenarios require a delicate calibration phase, often made even morechallenging by the scarcity and uncertainty of the observed data reported byofficial sources. In this context, Physics-Informed Neural Networks (PINNs), byembedding the knowledge of the differential model that governs the physicalphenomenon in the learning process, can effectively address the inverse andforward problem of data-driven learning and solving the corresponding epidemicproblem. In many circumstances, however, the spatial propagation of aninfectious disease is characterized by movements of individuals at differentscales governed by multiscale PDEs. This reflects the heterogeneity of a regionor territory in relation to the dynamics within cities and in neighboringzones. In presence of multiple scales, a direct application of PINNs generallyleads to poor results due to the multiscale nature of the differential model inthe loss function of the neural network. To allow the neural network to operateuniformly with respect to the small scales, it is desirable that the neuralnetwork satisfies an Asymptotic-Preservation (AP) property in the learningprocess. To this end, we consider a new class of AP Neural Networks (APNNs) formultiscale hyperbolic transport models of epidemic spread that, thanks to anappropriate AP formulation of the loss function, is capable to work uniformlyat the different scales of the system. A series of numerical tests fordifferent epidemic scenarios confirms the validity of the proposed approach,highlighting the importance of the AP property in the neural network whendealing with multiscale problems especially in presence of sparse and partiallyobserved systems.

Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread