mlsanityData to 7 Oct 2026

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

arXiv 2302.12337 ↗First posted 23 Feb 2023cs.LG math.APcited by 14

Not on the map. Why?

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.

Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). Builds on 26 of its 87 references that are in the map. First posted on arXiv 23 Feb 2023.

Builds on

26 of 87 references.

References · 26

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2015Automatic differentiation in machine learning: a surveycited by 846
  3. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  4. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  5. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  6. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  7. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  8. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  9. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  10. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  11. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  12. 2020Physics informed deep learning for computational elastodynamics without labeled datacited by 79
14 more
  1. 2019Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networkscited by 76
  2. 2018Deep Learning of Vortex Induced Vibrationscited by 74
  3. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  4. 2020A Physics-Informed Deep Learning Paradigm for Car-Following Modelscited by 40
  5. 2021A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimationcited by 37
  6. 2020Deep learning of free boundary and Stefan problemscited by 33
  7. 2021Physics-informed Learning for Identification and State Reconstruction of Traffic Densitycited by 13
  8. 2022PSO-PINN: Physics-Informed Neural Networks Trained with Particle Swarm Optimizationcited by 11
  9. 2020Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problemcited by 10
  10. 2020DeepSTCL: A Deep Spatio-temporal ConvLSTM for Travel Demand Predictioncited by 10
  11. 2017A Long Short-Term Memory Recurrent Neural Network Framework for Network Traffic Matrix Predictioncited by 8
  12. 2015Highway Traffic State Estimation with Mixed Connected and Conventional Vehiclescited by 7
  13. 2022PIAT: Physics Informed Adversarial Training for Solving Partial Differential Equationscited by 3
  14. 2016Prediction of functional ARMA processes with an application to traffic datacited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 5; 2025: 3; 2026: 3.

14 papers.

Sort:

  1. 2024Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KANcited by 25
  2. 2023Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problemscited by 12
  3. 2025Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Studycited by 11
  4. 2024Projected Neural Differential Equations for Learning Constrained Dynamicscited by 5
  5. 2023Operator learning for hyperbolic partial differential equationscited by 4
  6. 2025Learning robust parameter inference and density reconstruction in flyer plate impact experimentscited by 2
  7. 2023Incorporating Nonlocal Traffic Flow Model in Physics-informed Neural Networkscited by 1
  8. 2024Capturing Shock Waves by Relaxation Neural Networkscited by 1
  9. 2024Science based AI model certification for untrained operational environments with application in traffic state estimationcited by 0
  10. 2024Science based AI model certification for new operational environments with application in traffic state estimationcited by 0
  11. 2025Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systemscited by 0
  12. 2026Physics-Informed Teacher-Student Ensemble Learning for Traffic State Estimation with a Varying Speed Limit Scenariocited by 0
2 more
  1. 2026A Closed-loop, State-centric, Multi-agent Framework for Passenger Load Estimation from Heterogeneous Data Streamscited by 0
  2. 2026Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Lawscited by 0