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.

翻译：我们首次提出了一种针对紧致度量图上连续支撑概率分布的深度生成式建模框架。给定度量图上的源测度和目标测度，该方法将图嵌入光滑环境空间，通过神经半对偶参数化求解熵正则化Kantorovich问题，并将生成样本投影回原始图。我们研究两种嵌入几何结构：外蕴欧几里得实现与内蕴热带阿贝尔-雅可比嵌入（嵌入至雅可比环面）。两种情况下，生成器均通过构造天然具有图支撑。我们证明，在神经表达能力递增的联合极限下，学习到的生成器弱收敛于原始图测度之间的有效传输耦合。实验表明，在多种几何构型不同的图上，该方法与基于离散图最优传输的启发式传输基线持平或更优，且扩展性更佳。最后，我们通过在纽约市曼哈顿区一百万个Uber上车点数据上训练模型，验证了该方法在真实城市交通数据上的可扩展性。