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

arXiv: https://arxiv.org/abs/2302.12337 · HTML: https://mlsanity.com/paper/2302.12337/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 87 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [A Physics-Informed Deep Learning Paradigm for Car-Following Models](https://mlsanity.com/paper/2012.13376/) (2020)
- [A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation](https://mlsanity.com/paper/2106.03142/) (2021)
- [Deep learning of free boundary and Stefan problems](https://mlsanity.com/paper/2006.05311/) (2020)
- [Physics-informed Learning for Identification and State Reconstruction of Traffic Density](https://mlsanity.com/paper/2103.13852/) (2021)
- [PSO-PINN: Physics-Informed Neural Networks Trained with Particle Swarm Optimization](https://mlsanity.com/paper/2202.01943/) (2022)
- [Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem](https://mlsanity.com/paper/2001.05172/) (2020)
- [DeepSTCL: A Deep Spatio-temporal ConvLSTM for Travel Demand Prediction](https://mlsanity.com/paper/2009.00096/) (2020)
- [A Long Short-Term Memory Recurrent Neural Network Framework for Network Traffic Matrix Prediction](https://mlsanity.com/paper/1705.05690/) (2017)
- [Highway Traffic State Estimation with Mixed Connected and Conventional Vehicles](https://mlsanity.com/paper/1504.06879/) (2015)
- [PIAT: Physics Informed Adversarial Training for Solving Partial Differential Equations](https://mlsanity.com/paper/2207.06647/) (2022)
- [Prediction of functional ARMA processes with an application to traffic data](https://mlsanity.com/paper/1603.02049/) (2016)

## Built on by (most cited first, 14 of 14)

- [Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN](https://mlsanity.com/paper/2407.18373/) (2024)
- [Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems](https://mlsanity.com/paper/2308.07051/) (2023)
- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [Projected Neural Differential Equations for Learning Constrained Dynamics](https://mlsanity.com/paper/2410.23667/) (2024)
- [Learning robust parameter inference and density reconstruction in flyer plate impact experiments](https://mlsanity.com/paper/2506.23914/) (2025)
- [Incorporating Nonlocal Traffic Flow Model in Physics-informed Neural Networks](https://mlsanity.com/paper/2308.11818/) (2023)
- [Capturing Shock Waves by Relaxation Neural Networks](https://mlsanity.com/paper/2404.01163/) (2024)
- [Science based AI model certification for untrained operational environments with application in traffic state estimation](https://mlsanity.com/paper/2403.14093/) (2024)
- [Science based AI model certification for new operational environments with application in traffic state estimation](https://mlsanity.com/paper/2405.07893/) (2024)
- [Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systems](https://mlsanity.com/paper/2506.19609/) (2025)
- [Physics-Informed Teacher-Student Ensemble Learning for Traffic State Estimation with a Varying Speed Limit Scenario](https://mlsanity.com/paper/2605.11346/) (2026)
- [A Closed-loop, State-centric, Multi-agent Framework for Passenger Load Estimation from Heterogeneous Data Streams](https://mlsanity.com/paper/2605.19834/) (2026)
- [Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws](https://mlsanity.com/paper/2608.10389/) (2026)

## Abstract (arXiv)

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.