Self-adaptive equation embedded neural networks for traffic flow state estimation with sparse data

The data-driven approach in intelligent traffic systems has achieved successive breakthroughs, thanks to the ever-increasing volume of traffic data. Nonetheless, in practical scenarios, the collected data often contain some issues, e.g., missing values, significantly impacting the accuracy and efficiency of the algorithms. To enhance the precision of traffic estimation utilizing the sparse data, we have developed a physics-informed neural network (PINN) based algorithm in the line with the traffic flow theory and deep learning principles. In contrast to the conventional PINNs, our approach uniquely incorporates a self-adaptive macro model for mixed flow into the network's architecture, serving as an embedded source of physics information. With this algorithm, we can capture the dynamic behavior of an entire traffic flow including its spatiotemporal evolution with sparse traffic data such as initial and boundary value information. To realize the model's adaptability, we have revised the macro model by inverting its parameters and incorporating a data-driven nonlinear element, which simplifies the intricate macro model structure. The network's effectiveness has been validated through the experiments conducted on a mixed traffic flow system experiencing local agglomeration and real-world data, demonstrating its capability for precise traffic simulation, efficient traffic flow prediction, and interpretability. Our study offers a novel insight for data-driven traffic flow state estimation.