Optimization of continuous-flow over traffic networks with fundamental diagram constraints

TL;DR

This paper introduces a novel optimal transport model for traffic networks that incorporates fundamental diagram constraints, enabling congestion-aware flow optimization with proven existence, uniqueness, and scalable numerical methods.

Contribution

It extends classical optimal transport by integrating empirical fundamental diagrams into a convex, variational framework for dynamic traffic flow modeling.

Findings

The model guarantees mass conservation and convexity.

Existence and uniqueness of optimal flows are proven.

Numerical experiments demonstrate scalability to city-level networks.

Abstract

Optimal transport (OT) theory provides a principled framework for modeling mass movement in applications such as mobility, logistics, and economics. Classical formulations, however, generally ignore capacity limits that are intrinsic in applications, in particular in traffic flow problems. We address this limitation by incorporating fundamental diagrams into a dynamic continuous-flow OT model on graphs, thereby including empirical relations between local density and maximal flux. We adopt an Eulerian kinetic action on graphs that preserves displacement interpolation in direct analogy with the continuous theory. Momentum lives on edges and density on nodes, mirroring road-network semantics in which segments carry speed and intersections store mass. The resulting fundamental-diagram-constrained OT problem preserves mass conservation and admits a convex variational discretization, yielding optimal congestion-aware traffic flow over road networks. We establish the existence and uniqueness of the optimal flow with sources and sinks, and develop an efficient convex optimization method. Numerical studies begin with a single-lane line network and scale to a city-level road network.