Generative Modeling on Metric Graphs via Neural Optimal Transport

2026-06-15

AI Summary

Existing deep generative models struggle to handle continuous probability distributions supported on compact metric graphs. This work proposes a novel approach that embeds the metric graph into a smooth ambient space—accommodating both extrinsic Euclidean and intrinsic tropical Abel–Jacobi embeddings—and solves an entropy-regularized Kantorovich problem via neural semi-dual parameterization, subsequently projecting generated samples back onto the original graph. To the best of our knowledge, this is the first method enabling deep generative modeling of probability distributions with continuous support on metric graphs, with theoretical guarantees that the generator converges weakly to the true optimal transport coupling. Experiments demonstrate that the method matches or outperforms discrete graph-based optimal transport baselines across diverse geometric graph structures and successfully scales to million-scale Manhattan Uber pickup location data.

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.

Problem

• generative modeling
• metric graphs
• optimal transport
• probability distributions
• neural networks

Innovation

• Neural Optimal Transport
• Metric Graphs
• Generative Modeling
• Abel–Jacobi Embedding
• Entropic Kantorovich Problem