Generative Modeling on Metric Graphs via Neural Optimal Transport

Abstract:We introduce, to our knowledge, the first deep generative modeling framework for probability distributions continuously supported on compact metric graphs. Given source and target measures on a metric graph, our method embeds the graph into a smooth ambient space, solves an entropic Kantorovich problem via a neural semidual parameterization, and projects generated samples back onto the original graph. We study two embedded geometries: an extrinsic Euclidean realization and the intrinsic tropical Abel--Jacobi embedding into the Jacobian torus. In both cases, the resulting generator is graph-supported by construction. We prove that, in the joint limit of increasing neural expressivity, the learned generator converges weakly to a valid transport coupling between the original graph measures. Empirically, across a range of geometrically distinct graphs, our method matches or improves upon heuristic transport baselines based on discrete graph OT, while scaling more favorably. Finally, we demonstrate scalability on real-world urban mobility data by training our model on one million Uber pickup locations in Manhattan, New York City.

Machine Learning (stat.ML); Machine Learning (cs.LG); Methodology (stat.ME)

arXiv:2606.16273 [stat.ML]

(or arXiv:2606.16273v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2606.16273 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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