Learning to learn the macroscopic fundamental diagram using physics-informed and model agnostic machine learning

• A model-agnostic meta-learning framework that leverages data from multiple cities to estimate Macroscopic Fundamental Diagrams (MFDs) in networks with scarce data from loop detectors. • A Multi-Task Physics-Informed Neural Network (MTPINN) that jointly predicts average flow and critical occupancy and obtains a MFD shape through physics-informed regularisation. • Results show that coupling MAML with MTPINN substantially improves MFD estimation under data scarcity, reducing average MSE and remaining effective with as few as 10 loop detectors per city. • The proposed meta-learning approach generalises robustly across 20 heterogeneous cities from the UTD19 dataset and outperforms conventional neural networks, bi-parabolic baselines, and standard transfer learning. The Macroscopic Fundamental Diagram is a popular tool used to describe traffic dynamics in an aggregated way, with applications ranging from traffic control to incident analysis. However, estimating the MFD for a given network requires large numbers of loop detectors, which is not always available in practise. This article proposes a framework to alleviate the data scarcity challenge harnessing Meta-Learning, a subcategory of Machine Learning that trains models to understand and adapt to new tasks on their own. We use Meta-Learning to identify and exploit transferable patterns from data-rich cities to cities where not enough data is available to estimate the MFD. The developed model is trained and tested by leveraging data from multiple cities and exploiting it to model the MFD of other cities with different shares of detectors and topological structures. The proposed Meta-Learning framework is applied to an ad-hoc Multi-Task Physics-Informed Neural Network, specifically designed to estimate the MFD. Results show an average MAE improvement in flow prediction of around 50% across cities (depending on the subset of loop detectors tested). The Meta-Learning framework thus successfully generalises across diverse urban settings and improves performance on cities with limited data, demonstrating the potential of using Meta-Learning when a limited number of detectors is available. We directly test this assumption by applying the Meta-Learning outputs to unseen cities to simulate a real-life application scenario and the wide applicability of the proposed methodology. Finally, the proposed framework is validated against traditional Transfer Learning approaches and tested with FitFun, a model for FD estimation from the literature, to prove its transferability.