标题： 通过邻近分裂方法约束的动态最优运输的基本图 显示英文标题

标题： Fundamental diagram constrained dynamic optimal transport via proximal splitting methods

摘要： 最优运输最近被提出作为一种工具，用于建模和高效解决各种流问题，例如起讫点问题和多商品流问题。 尽管该框架已被证明对许多大规模流问题有效，但其公式通常缺乏常见交通模型中使用的动态特性，例如Lighthill-Whitham-Richards模型。 在本工作中，我们提出了一种包含由基本图指定的动态约束的最优运输框架，用于建模宏观交通流。 该问题被表述为动态最优运输的一个凸变体，具有额外的非线性时空不等式约束，这些约束模仿了交通理论中的基本图。 该约束在可接受流量上施加了一个与密度相关的上限，捕捉了流量饱和和拥堵效应，从而为动力学优化留出了空间。 该公式遵循Benamou-Brenier运输原理，其中动能在密度和动量场上进行优化，同时满足质量守恒定律。 我们开发了邻近分裂方法，即Douglas-Rachford和Chambolle-Pock算法，这些算法利用了约束集的可分离结构，并且只需要简单的邻近操作，可以适应额外的（时变）空间限制或障碍物。 数值实验说明了约束对运输行为的影响，包括拥堵感知的扩散、重新路由和收敛。 该框架建立了最优运输与宏观交通流理论之间的联系，并提供了一个可扩展的变分工具，用于建模受拥堵限制（或饱和感知）的Wasserstein梯度流。 显示英文摘要

摘要： Optimal transport has recently been brought forward as a tool for modeling and efficiently solving a variety of flow problems, such as origin-destination problems and multi-commodity flow problems. Although the framework has shown to be effective for many large scale flow problems, the formulations typically lack dynamic properties used in common traffic models, such as the Lighthill-Whitham-Richards model. In this work, we propose an optimal transport framework that includes dynamic constraints specified by the fundamental diagram for modeling macroscopic traffic flow. The problem is cast as a convex variant of dynamic optimal transport, with additional nonlinear temporal-spatial inequality constraints of momentum, modeled after the fundamental diagram from traffic theory. This constraint imposes a density-dependent upper bound on the admissible flux, capturing flow saturation and congestion effects, and thus leaves space for kinetic optimization. The formulation follows the Benamou-Brenier transportation rationale, whereby kinetic energy over density and momentum fields is optimized subject to the mass conservation law. We develop proximal splitting methods, namely the Douglas-Rachford and Chambolle-Pock algorithms, which exploit the separable structure of the constraint set and require only simple proximal operations, and can accommodate additional (time-varying) spatial restrictions or obstacles. Numerical experiments illustrate the impact of the constraint on transport behavior, including congestion-aware spreading, rerouting, and convergence. The framework establishes a connection between optimal transport and macroscopic traffic flow theory and provides a scalable, variational tool for modeling congestion-constricted (or saturation-aware) Wasserstein gradient flow.