Accurate traffic flow estimation and prediction are critical for the efficient management of transportation systems, particularly under increasing urbanization. Traditional methods relying on static sensors often suffer from limited spatial coverage, while probe vehicles provide richer, albeit sparse and irregular data. This work introduces ON-Traffic, a novel deep operator Network and a receding horizon learning-based framework tailored for online estimation of spatio-temporal traffic state along with quantified uncertainty by using measurements from moving probe vehicles and downstream boundary inputs. Our framework is evaluated in both numerical and simulation datasets, showcasing its ability to handle irregular, sparse input data, adapt to time-shifted scenarios, and provide well-calibrated uncertainty estimates. The results demonstrate that the model captures complex traffic phenomena, including shockwaves and congestion propagation, while maintaining robustness to noise and sensor dropout. These advancements present a significant step toward online, adaptive traffic management systems.

We describe a recurrent neural network (RNN) based architecture to learn the flow function of a causal, time-invariant and continuous-time control system from trajectory data. By restricting the class of control inputs to piecewise constant functions, we show that learning the flow function is equivalent to learning the input-to-state map of a discrete-time dynamical system. This motivates the use of an RNN together with encoder and decoder networks which map the state of the system to the hidden state of the RNN and back. We show that the proposed architecture is able to approximate the flow function by exploiting the system's causality and time-invariance. The output of the learned flow function model can be queried at any time instant. We experimentally validate the proposed method using models of the Van der Pol and FitzHugh Nagumo oscillators. In both cases, the results demonstrate that the architecture is able to closely reproduce the trajectories of these two systems. For the Van der Pol oscillator, we further show that the trained model generalises to the system's response with a prolonged prediction time horizon as well as control inputs outside the training distribution. For the FitzHugh-Nagumo oscillator, we show that the model accurately captures the input-dependent phenomena of excitability.

Designing Luenberger observers for nonlinear systems involves the challenging task of transforming the state to an alternate coordinate system, possibly of higher dimensions, where the system is asymptotically stable and linear up to output injection. The observer then estimates the system's state in the original coordinates by inverting the transformation map. However, finding a suitable injective transformation whose inverse can be derived remains a primary challenge for general nonlinear systems. We propose a novel approach that uses supervised physics-informed neural networks to approximate both the transformation and its inverse. Our method exhibits superior generalization capabilities to contemporary methods and demonstrates robustness to both neural network's approximation errors and system uncertainties.