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Since its introduction in 2017, physics-informed deep learning (PIDL) hasgarnered growing popularity in understanding the evolution of systems governedby physical laws in terms of partial differential equations (PDEs). However,empirical evidence points to the limitations of PIDL for learning certain typesof PDEs. In this paper, we (a) present the challenges in training PIDLarchitecture, (b) contrast the performance of PIDL architecture in learning afirst order scalar hyperbolic conservation law and its parabolic counterpart,(c) investigate the effect of training data sampling, which corresponds tovarious sensing scenarios in traffic networks, (d) comment on the implicationsof PIDL limitations for traffic flow estimation and prediction in practice.Detailed in the case study, we present the contradistinction in PIDL resultsbetween learning the traffic flow model (LWR PDE) and its variation withdiffusion. The outcome indicates that PIDL experiences significant challengesin learning the hyperbolic LWR equation due to the non-smoothness of itssolution. On the other hand, the architecture with parabolic PDE, augmentedwith the diffusion term, leads to the successful reassembly of the density dataeven with the shockwaves present.

On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models