Optimizing traffic on a network is a relevant problem in situations where traffic congestion has a big impact on transmission performance. This can be formalized as a computationally-hard constrained optimization problem where interactions are non-local and a global optimization is required.
We investigate this problem by adopting approaches that combine insights from statistical physics and an optimal transport theory.

We tackle this challenge by first developing theoretical models to lay the foundations of these approaches.

We consider realistic scenarios like multi-commodity frameworks, where flows of different types compete for a shared usage of network infrastructure. In transportation networks, these can represent passengers of different types, e.g. with different demands and needs. The problem can be model in different ways. We investigate various approaches, like the standard optimal transport scenario where a unique global cost function is minimized, taking the perspective of a network manager.  Alternatively, one can frame this as a bi-level optimization problem, where a second type of cost is considered in a hierarchical way. The lower-level cost can represent for instance passengers' selfish routing, which contrasts with the need of a central network manager, the top-level cost.

We analyze how multiple transportation modes can be combined, as in a multilayer network, where the cost and weight to travel on different network edges varies. With this, we can assess how passengers flows can be encouraged towards a certain transportation mode, e.g. a more sustainable mode as rails, while keeping the trajectories optimized for metrics like path length.

In addition, we consider principled ways to add constraints into the framework, in a physics-inspired constrained optimization formalism that flexibly adapts to various types of constraints (linear, non-linear, local, global).

We apply these models on a variety of concrete applications, ranging from transport in urban transportation networks to machine learning tasks like image classification or automatic extraction of network-like objects from images.

The numerical implementations of our models are available open source online.