LEARNING THE SHAPE OF DEMAND: A GEOMETRIC FRAMEWORK FOR REAL-TIME SHARED MOBILITY

Kurzfassung

The proliferation of shared and automated transport services has challenged long-standing paradigms in urban mobility. As cities evolve to support flexible, sustainable, and demand-responsive transit, new computational and infrastructural concepts are needed to coordinate users, vehicles, and physical space in real-time. At the center of this transformation is the concept of “virtual stops”, dynamic data-driven access point where passengers engage with on-demand mobility services. Virtual stops offer the potential to bridge the rigidity of fixed infrastructure with the fluidity of user demand, enabling more adaptive and efficient transport systems. Recent studies have made important progress toward operationalizing virtual stops in practice. Rule-based methods proposed by Harmann et al. (2022, 2023) and infrastructure-aware platforms like KoKoVi (Touko & Rummel, 2023) define and evaluate virtual stop locations based on static spatial features such as intersections, parking bays, or road furniture and regulatory or safety constraints. Parallel efforts by Hub et al. (2023) emphasize user experience, introducing augmented reality-based interfaces to help riders locate stops and vehicles, particularly in unfamiliar or unmarked environments. While these contributions are essential for ensuring feasibility and usability, they also share a common assumption: that virtual stops must be externally defined, selected from a set of static spatial candidates and then filtered based on constraints. Demand is considered retrospectively, used to justify stop deployment, but not to generate or shape it. As a result, these systems risk decoupling virtual stop design from the dynamic, spatiotemporal patterns of urban mobility itself. To address this gap, this research proposes a change of paradigm. Drawing on current advances in geometric deep learning, manifold learning and topological data analysis, we propose a framework in which virtualstops and user assignments are inferred as emergentstructures within a dynamic spatiotemporal system (Pham et al., 2025; Hofer et al., 2017, Berry & Sauer, 2019; Chakraborty et al., 2020). More specifically, we model urban mobility demand as a continuous field over space and time, learned from historical and real-time origin-destination (OD) data. This field induces a Riemannian geometry over the urban space, where regions of high curvature correspond to zones of converging user intent. These high-intensity regions act as natural attractorsforshared transport and are interpreted as emergent virtual stop locations, adapting dynamically to shifts in demand across time and space. To represent not only pairwise, but also group-level interactions among users, stops, and vehicles, we use combinatorial complexes (CC), a generalization of graphs that supports higher-order relationships and temporal evolution (Battiston et al., 2021). These structures enable a more expressive encoding of ride-sharing assignments and multi-user coordination, moving beyond traditional graph-based methods. On this evolving structure, we define assignment signals – probabilistic or flow-based relationships between users, vehicles, and virtual stops – using Hodge decomposition, a technique from algebraic topology that separates these flows into three meaningful components: a gradient term capturing global coordination, a curl term representing localized inefficiencies or routing loops, and a harmonic term indicating structural or topological bottlenecks (Aoki et al., 2022; Kan & López, 2022). These components are both interpretable, but also learnable,serving as objectives and constraints in our predictive model. Another key advantage of this approach is its ability to cluster demand naturally, increasing vehicle occupancy and reducing the number of vehicles required to satisfy user needs (Hermann et al., 2024). This is particularly important for improving pooling efficiency and achieving scalable, sustainable operations. The framework is evaluated using both real-world and synthetic datasets and benchmark against rule-based and graph-based baselines. Results might show that emergent virtual stops closely align with demand hotspots and adjust fluidly as conditions change and lead to measurable improvements in waiting times, vehicle utilization, and route efficiency. In addition, at the topological level, we analyze the decomposition of assignment flows, quantifying reductions in cyclic routing and clearer detection of structurally constrained areas via harmonic energy. To conclude, this work proposes a new perspective for shared mobility systems, one that does not merely optimize within existing constraints, but learnsstructure from collective behavior, and coordinates dynamically as user requests change and evolve.