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STDEN: Towards Physics-Guided Neural Networks for Traffic Flow Prediction

Jiahao Ji, Jingyuan Wang, Zhe Jiang, Jiawei Jiang, Hu Zhang

TL;DR

This work addresses the tension between physics-based interpretability and data-driven accuracy in traffic flow prediction by introducing STDEN, a physics-guided neural network that learns a latent Traffic Potential Energy Field (PEF) and evolves it with a Spatio-Temporal Differential Equation Network. The model couples an encoder–DE–decoder architecture with a Physics-Guided traffic Flow Modeling (PGFM) component, deriving a PEF continuity equation $\frac{\partial \mathbf{z}}{\partial t} = - \boldsymbol{\phi} \odot (\alpha \Delta \mathbf{z})$ and implementing it as a residual Graph Convolutional Network within a neural ODE framework. Empirical results on three Beijing sub-networks show substantial improvements over state-of-the-art baselines across multiple horizons, and qualitative analyses demonstrate that the learned PEF captures the physical mechanism driving urban traffic, offering interpretability. The proposed framework provides a general, physics-informed approach that could be extended to other spatio-temporal systems, such as weather forecasting or epidemic modeling, where dynamics are governed by diffusive transport on networks.

Abstract

High-performance traffic flow prediction model designing, a core technology of Intelligent Transportation System, is a long-standing but still challenging task for industrial and academic communities. The lack of integration between physical principles and data-driven models is an important reason for limiting the development of this field. In the literature, physics-based methods can usually provide a clear interpretation of the dynamic process of traffic flow systems but are with limited accuracy, while data-driven methods, especially deep learning with black-box structures, can achieve improved performance but can not be fully trusted due to lack of a reasonable physical basis. To bridge the gap between purely data-driven and physics-driven approaches, we propose a physics-guided deep learning model named Spatio-Temporal Differential Equation Network (STDEN), which casts the physical mechanism of traffic flow dynamics into a deep neural network framework. Specifically, we assume the traffic flow on road networks is driven by a latent potential energy field (like water flows are driven by the gravity field), and model the spatio-temporal dynamic process of the potential energy field as a differential equation network. STDEN absorbs both the performance advantage of data-driven models and the interpretability of physics-based models, so is named a physics-guided prediction model. Experiments on three real-world traffic datasets in Beijing show that our model outperforms state-of-the-art baselines by a significant margin. A case study further verifies that STDEN can capture the mechanism of urban traffic and generate accurate predictions with physical meaning. The proposed framework of differential equation network modeling may also cast light on other similar applications.

STDEN: Towards Physics-Guided Neural Networks for Traffic Flow Prediction

TL;DR

This work addresses the tension between physics-based interpretability and data-driven accuracy in traffic flow prediction by introducing STDEN, a physics-guided neural network that learns a latent Traffic Potential Energy Field (PEF) and evolves it with a Spatio-Temporal Differential Equation Network. The model couples an encoder–DE–decoder architecture with a Physics-Guided traffic Flow Modeling (PGFM) component, deriving a PEF continuity equation and implementing it as a residual Graph Convolutional Network within a neural ODE framework. Empirical results on three Beijing sub-networks show substantial improvements over state-of-the-art baselines across multiple horizons, and qualitative analyses demonstrate that the learned PEF captures the physical mechanism driving urban traffic, offering interpretability. The proposed framework provides a general, physics-informed approach that could be extended to other spatio-temporal systems, such as weather forecasting or epidemic modeling, where dynamics are governed by diffusive transport on networks.

Abstract

High-performance traffic flow prediction model designing, a core technology of Intelligent Transportation System, is a long-standing but still challenging task for industrial and academic communities. The lack of integration between physical principles and data-driven models is an important reason for limiting the development of this field. In the literature, physics-based methods can usually provide a clear interpretation of the dynamic process of traffic flow systems but are with limited accuracy, while data-driven methods, especially deep learning with black-box structures, can achieve improved performance but can not be fully trusted due to lack of a reasonable physical basis. To bridge the gap between purely data-driven and physics-driven approaches, we propose a physics-guided deep learning model named Spatio-Temporal Differential Equation Network (STDEN), which casts the physical mechanism of traffic flow dynamics into a deep neural network framework. Specifically, we assume the traffic flow on road networks is driven by a latent potential energy field (like water flows are driven by the gravity field), and model the spatio-temporal dynamic process of the potential energy field as a differential equation network. STDEN absorbs both the performance advantage of data-driven models and the interpretability of physics-based models, so is named a physics-guided prediction model. Experiments on three real-world traffic datasets in Beijing show that our model outperforms state-of-the-art baselines by a significant margin. A case study further verifies that STDEN can capture the mechanism of urban traffic and generate accurate predictions with physical meaning. The proposed framework of differential equation network modeling may also cast light on other similar applications.
Paper Structure (20 sections, 16 equations, 5 figures, 3 tables)

This paper contains 20 sections, 16 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Definition
  4. Physics-Guided Traffic Flow Modeling
  5. The Model Framework
  6. Physics-Based Continuity Equation
  7. Continuity Equation for Traffic PEF
  8. Spatio-Temporal Differential Equation Network
  9. Differential Equation Network
  10. DE-Net for Traffic Potential Energy Fields
  11. The Model Training
  12. Experiments
  13. Datasets
  14. Experimental Settings
  15. Performance Comparison
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Illustration of traffic flow and potential energy field defined on road networks. The arrow indicates the traffic flow direction while its size denotes the flow volume. In panel (b), higher cylinder means more potential energy, and the traffic flow volume (arrow size) increases with the energy gradients between adjacent nodes.
  • Figure 2: Overview of the proposed physics-guided model. STDEN: Spatio-Temporal Differential Equation Network. PGFM: Physics-Guided traffic Flow Modeling. PF-Trans: transformation between the PEF and traffic flow (Eq. \ref{['eq:potn_flow']}).
  • Figure 3: Evaluation on potential energy field differential equation over all datasets.
  • Figure 4: Computation overhead vs. Prediction accuracy. NFE denotes number of function evaluation.
  • Figure 5: Visualization of the learned potential energy fields and real traffic flow on GT-221 dataset. The heat map represents the potential energy fields, while the arrows denote traffic flow with its volume reflected by the color and the arrow size. The potential energy fields learned by STDEN can interpret the traffic flow.

Theorems & Definitions (4)

  • Definition 1: Road Network
  • Definition 2: Traffic Flow
  • Definition 3: Traffic Potential Energy Field (PEF)
  • Definition 4: Flow-sequence and PEF-sequence