# On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models Source: [On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models]() Collection index: [llms.txt]() Data updated: 2025-10-27T05:26:34.657Z Authors: Archie J. Huang, Shaurya Agarwal Published: 2023-02-23T21:08:02.000Z ## Abstract Since its introduction in 2017, physics-informed deep learning (PIDL) has garnered growing popularity in understanding the evolution of systems governed by physical laws in terms of partial differential equations (PDEs). However, empirical evidence points to the limitations of PIDL for learning certain types of PDEs. In this paper, we (a) present the challenges in training PIDL architecture, (b) contrast the performance of PIDL architecture in learning a first order scalar hyperbolic conservation law and its parabolic counterpart, (c) investigate the effect of training data sampling, which corresponds to various sensing scenarios in traffic networks, (d) comment on the implications of PIDL limitations for traffic flow estimation and prediction in practice. Detailed in the case study, we present the contradistinction in PIDL results between learning the traffic flow model (LWR PDE) and its variation with diffusion. The outcome indicates that PIDL experiences significant challenges in learning the hyperbolic LWR equation due to the non-smoothness of its solution. On the other hand, the architecture with parabolic PDE, augmented with the diffusion term, leads to the successful reassembly of the density data even with the shockwaves present. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Traffic State Estimation on I-24 MOTION 1.0 (20221122)](): Traffic State Estimation; I-24 MOTION 1.0 (20221122); Mean Relative L2 Error: 14.88. - [Traffic State Estimation on I-24 MOTION 1.0 (20221123)](): Traffic State Estimation; I-24 MOTION 1.0 (20221123); Mean relative L2 Error (%): 8.75. - [Traffic State Estimation on I-24 MOTION 1.0 (20221202)](): Traffic State Estimation; I-24 MOTION 1.0 (20221202); Mean Relative L2 Error (%): 14.36. - [Traffic trajectory estimation on I-24 MOTION](): Traffic trajectory estimation; I-24 MOTION; L2 Error: 16.81. - [Traffic State Estimation on I-24 MOTION 1.0 (20221121)](): Traffic State Estimation; I-24 MOTION 1.0 (20221121); Mean Relative L2 Error: 14.26. - [Traffic State Estimation on I-24 MOTION 1.0 (20221129)](): Traffic State Estimation; I-24 MOTION 1.0 (20221129); Mean relative L2 error (%): 19.26. ## Original sources and implementation [Original paper]() [PDF]()