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Neural Networks Enabled Discovery On the Higher-Order Nonlinear Partial Differential Equation of Traffic Dynamics

Zihang Wei, Yunlong Zhang, Chenxi Liu, Yang Zhou

TL;DR

The paper addresses the challenge of discovering high-order nonlinear PDEs that govern traffic network dynamics from noisy, sparsely sampled sensor data. It introduces TRAFFIC-PDE-LEARN, a data-driven framework that (i) reconstructs dense spatiotemporal traffic states via neural networks modeling spatiotemporal heterogeneous fundamental diagrams, (ii) computes derivatives with automatic differentiation, and (iii) identifies a sparse, Koopman-based linear representation of the hidden PDE. The approach yields a validated second-order nonlinear PDE for occupancy, which is then used to predict traffic flow over 3–15 minute horizons, outperforming CTM, LSTM, and RNN baselines. The results demonstrate the practical potential for data-driven PDE discovery to enhance intelligent transportation systems, while acknowledging limitations from sensor spacing and suggesting future validation with denser data from connected vehicles.

Abstract

Modeling the traffic dynamics is essential for understanding and predicting the traffic spatiotemporal evolution. However, deriving the partial differential equation (PDE) models that capture these dynamics is challenging due to their potential high order property and nonlinearity. In this paper, we introduce a novel deep learning framework, "TRAFFIC-PDE-LEARN", designed to discover hidden PDE models of traffic network dynamics directly from measurement data. By harnessing the power of the neural network to approximate a spatiotemporal fundamental diagram that facilitates smooth estimation of partial derivatives with low-resolution loop detector data. Furthermore, the use of automatic differentiation enables efficient computation of the necessary partial derivatives through the chain and product rules, while sparse regression techniques facilitate the precise identification of physically interpretable PDE components. Tested on data from a real-world traffic network, our model demonstrates that the underlying PDEs governing traffic dynamics are both high-order and nonlinear. By leveraging the learned dynamics for prediction purposes, the results underscore the effectiveness of our approach and its potential to advance intelligent transportation systems.

Neural Networks Enabled Discovery On the Higher-Order Nonlinear Partial Differential Equation of Traffic Dynamics

TL;DR

The paper addresses the challenge of discovering high-order nonlinear PDEs that govern traffic network dynamics from noisy, sparsely sampled sensor data. It introduces TRAFFIC-PDE-LEARN, a data-driven framework that (i) reconstructs dense spatiotemporal traffic states via neural networks modeling spatiotemporal heterogeneous fundamental diagrams, (ii) computes derivatives with automatic differentiation, and (iii) identifies a sparse, Koopman-based linear representation of the hidden PDE. The approach yields a validated second-order nonlinear PDE for occupancy, which is then used to predict traffic flow over 3–15 minute horizons, outperforming CTM, LSTM, and RNN baselines. The results demonstrate the practical potential for data-driven PDE discovery to enhance intelligent transportation systems, while acknowledging limitations from sensor spacing and suggesting future validation with denser data from connected vehicles.

Abstract

Modeling the traffic dynamics is essential for understanding and predicting the traffic spatiotemporal evolution. However, deriving the partial differential equation (PDE) models that capture these dynamics is challenging due to their potential high order property and nonlinearity. In this paper, we introduce a novel deep learning framework, "TRAFFIC-PDE-LEARN", designed to discover hidden PDE models of traffic network dynamics directly from measurement data. By harnessing the power of the neural network to approximate a spatiotemporal fundamental diagram that facilitates smooth estimation of partial derivatives with low-resolution loop detector data. Furthermore, the use of automatic differentiation enables efficient computation of the necessary partial derivatives through the chain and product rules, while sparse regression techniques facilitate the precise identification of physically interpretable PDE components. Tested on data from a real-world traffic network, our model demonstrates that the underlying PDEs governing traffic dynamics are both high-order and nonlinear. By leveraging the learned dynamics for prediction purposes, the results underscore the effectiveness of our approach and its potential to advance intelligent transportation systems.
Paper Structure (13 sections, 30 equations, 10 figures, 2 tables)

This paper contains 13 sections, 30 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Description and Framework
  3. Methodology
  4. Neural Networks Enabled Spatiotemporal Heterogeneous Fundamental Diagrams Approximation
  5. Computation of the Partial Derivatives of $\hat{o}$, $\hat{q}$, and $\hat{v}$
  6. Hidden PDE Discovery of Traffic Network Dynamics by Koopman Theory
  7. Holistic Design of the Proposed "TRAFFIC-PDE-LEARN" Model
  8. Training Process of the Proposed Model
  9. Experimental Results
  10. Data Preparation
  11. Training
  12. Traffic Flow Prediction based on the Discovered PDE Model
  13. Conclusion

Figures (10)

  • Figure 1: High-Level Design of the Proposed Model's Framework
  • Figure 2: (a) Structure of $\hat{o}$, (b) Structure of $q$, and (c) Structure of $v$
  • Figure 3: Holistic Design of the Proposed “TRAFFIC-PDE-LEARN” Model
  • Figure 4: Locations of the Traffic Sensors, and the Corresponding Noisy and Sparse Traffic Measurement
  • Figure 5: Loss Functions' Values during Burn-in, Main, and Refinement Steps.
  • ...and 5 more figures