Problem

The increasing demand for efficient and sustainable urban mobility, driven by factors like autonomous vehicles and UN Sustainable Development Goals, necessitates accurate real-time traffic density estimation. Traditional sensor-based methods are costly, lack scalability, and struggle with real-time adaptability. While Physics-Informed Neural Networks (PINNs) have shown promise for traffic state estimation, combining physics laws with data, their conventional training methods are computationally intensive and slow. This slowness prevents their application in real-time scenarios where continuous, up-to-date estimations are crucial, especially when dealing with time-varying system dynamics or model mismatches. The core challenge is to develop a causal, real-time algorithm that continuously provides accurate traffic density estimates from probe vehicle data, remains robust to model errors, and maintains accuracy despite varying training times and increasing data volumes.

Method

The paper proposes a novel framework for online traffic density estimation using adapted Physics-Informed Neural Networks. To overcome the limitations of traditional PINN training in a real-time context, the methodology addresses three key challenges: 1) Constant Neural Network Size: A maximum look-back time δ_d is introduced, creating a moving-time window for data. This ensures that the PINN's inference domain size remains constant, preventing a continuous increase in network size and maintaining accuracy. 2) Constant Training Time: Instead of training from scratch at each iteration, the model reuses parameters from the previous training as a warm-up. This is achieved by normalizing the time input and applying a calculated bias shift to the first layer, significantly reducing training time and retaining valuable historical information. 3) Robustness to Model Errors: The framework incorporates real-time learning of time-varying physical parameters, such as the free flow velocity (v_f), by fitting velocity measurements within a defined window (δ_v). This allows the PINN to adapt to changing traffic conditions (e.g., due to accidents or weather) and reduce the impact of model mismatch on estimation accuracy. The overall optimization problem combines data-driven loss (ℼ_ρ) from probe vehicle measurements with physics-informed loss (ℼ_˕ϕ) representing the underlying traffic flow PDEs.

Results

Numerical experiments were conducted in two settings: a Greenshield model and high-fidelity SUMO simulations. 1) Greenshield Model Comparison: When the model has perfect knowledge (no mismatch), a classical open-loop observer generally outperforms the PINN. However, in scenarios with significant model mismatch (e.g., varying free-flow velocity v_f), the PINN demonstrates superior performance by effectively adapting to the changing dynamics, achieving a strictly lower error. With low model error, both methods showed comparable performance over time. 2) Impact of Training Time: The study revealed a critical trade-off in the online setting. While offline models typically improve with more training iterations, the online PINN's error starts increasing when the number of epochs (e_t) exceeds approximately 300, due to the increasing irrelevance of older data caused by longer training times (δ_t). This highlights the necessity of balancing training intensity with real-time data recency. 3) SUMO Simulation: In a more realistic scenario with an unknown high-fidelity dynamical model (SUMO), the PINN successfully reproduced traffic characteristics. The Current Estimation Error (CEE) stabilized around 0.01 after an initial convergence period. Occasional spikes in error were observed due to data discontinuities or lack of probe vehicle data in certain road segments, which led to "erroneous green 'tails'" that subsequently disappeared upon new probe vehicle measurements.

Implications

The proposed online PINN framework offers a robust and adaptable solution for real-time traffic density estimation, particularly valuable for autonomous transport systems and traffic control strategies. Its ability to handle time-varying models and unknown high-fidelity dynamics, outperforming traditional methods in model mismatch scenarios, represents a significant step forward. Future research should focus on accelerating PINN convergence to minimize training time impact, incorporating uncertainty quantification to highlight reliability, and enhancing network adaptability through dynamic weighting of probe vehicle measurements.