Potential failures of physics-informed machine learning in traffic flow modeling: theoretical and experimental analysis

Potential failures of physics-informed machine learning in traffic flow modeling: theoretical and experimental analysis Yuan-Zheng Lei a, Yaobang Gong a, Dianwei Chen a, Yao Cheng b, Xianfeng Terry Yang* a a University of Maryland, College Park, MD 20742, United States b Florida Atlantic University, Boca Raton, FL 33431, United StatesAbstract This study investigates why physics-informed machine learning (PIML) may fail when it comes to macroscopic traffic flow modeling. We define failure as the case where a PIML model underperforms both its purely data-driven and purely physics-based counterparts by a given threshold. Our analysis shows that physics residuals themselves do not inherently hinder the optimization of the loss function, which is a main reason responsible for the failure of the PIML model in other fields. Instead, successful parameter updates require both machine-learning and physics gradients to form acute angles with the true gradient. Our experiment shows that this condition may be hard to achieve for PIML under a general low-resolution loop dataset. In particular, when the traffic data resolution is low, a neural network cannot accurately approximate density and speed, causing the constructed physics residuals, already affected by discrete sampling and temporal averaging, to lose their ability to reflect the actual PDE dynamics. This degradation can directly lead to PIML failure. From a theoretical standpoint, we show that although the exact solutions of the LWR and ARZ models are weak solutions, for piecewise C k initial data and under mild conditions, the solutions remain C k on the complement of the shock set over finite time, with only finitely many shock waves, where C k refers to k times continuously differentiable. Since the shock set has Lebesgue measure zero, the probability of a detector measurement or auxiliary collocation point lying exactly on a discontinuity is essentially zero; asymptotically, every auxiliary point admits a sufficiently small smooth neighborhood where the physics residual is well-defined and valid. Consequently, the well-known limitation that MLPs cannot exactly represent non-smooth functions does not materially affect our setting, as the residual evaluation almost always occurs in smooth regions. We also investigate the error lower bounds of the MSE of physics residuals for PIML models under high-resolution data. We prove that higher-order models like ARZ possess strictly larger consistency error lower bounds than lower-order models like LWR under mild conditions.