A physics-informed machine learning framework for speed-flow prediction: Integrating an S-shaped traffic stream model with deep learning models

In real-world situations, data insufficiency and missingness could significantly compromise the accuracy and reliability of traffic state prediction models. To address the challenge, this study incorporates prior knowledge of the fundamental diagram—which defines the relationship between flow (q) and speed (v)—into a Physics-Informed Machine Learning (PIML) framework to tackle the network-wide speed-flow prediction problem. The PIML model integrates Multi-Graph Convolutional Networks (MGCNs) and Long Short-Term Memory (LSTM) neural networks within a unified Computational Graph (CG) to capture the spatiotemporal dependencies of traffic states across sensor networks. To enhance the interpretability of the learning results, a calibratable S-shaped three-parameter (S3) traffic stream model is embedded into the PIML framework to regulate the relationships between key traffic variables, including speed, density, and flow, ensuring that the estimates and predictions are within a reasonable range satisfying traffic flow theories. The proposed model is validated using data from the Caltrans Performance Measurement System (PeMS), demonstrating superior performance over benchmark models on test datasets and exhibiting stronger generalization capabilities, even with limited data. The inclusion of the S-shaped traffic stream model not only improves predictive performance under conditions of incomplete or sparse data but also ensures inherent consistency with established traffic flow physics. The PIML model is particularly effective for inferring unobservable traffic flow from speed observations, making it a valuable tool for addressing corrupted and missing data points in traffic engineering applications.