The automatic segmentation of medical images plays an important role in diagnosis and therapy. Deep convolutional neural networks (CNN) represent the state of the art, but have limitations, particularly on the plausibility of the generated segmentations. Our hypothesis is that the improvement of segmentations will come from the addition of external information, via medical knowledge for example, and auxiliary tasks, such as registration, which will guide and constrain the segmentation. On the other hand, the uninterpretable nature of CNN hinders their use in the medical field. If there are explicability methods for classification, everything remains to be done for segmentation. We will aim to develop such methods, in order to understand the mechanisms underlying the addition of knowledge and tasks. Although our developments will be generic, we will target use cases to demonstrate the impact of the results on clinical practice.

The central theme of this project lies in the area of control theory and partial differential equations (in particular Hamilton-Jacobi equations), posed on stratified structures and networks. These equations appear very naturally in several applications like traffic flow modeling, energy management in smart-grids networks or sea-land trajectories with different dynamics. These control problems can be studied within the framework of Hamilton Jacobi equations theory. Recently, significant results have been obtained, leading to a good understanding of the notion of viscosity solutions (in particular the questions of existence and uniqueness) on some specific stratified structures. This base of results will be further developed in different directions. It will first be necessary to complete the analysis for more general problems under weaker hypotheses than the one used so far (nature of the stratification, hypotheses on the Hamiltonians, ...). On the other hand, it is necessary to use the already existing base to advance research in other active areas such as homogenization or mean field games. Moreover, all of the theoretical results will be used to achieve progress in the modeling and numerical resolution of some control problems on stratified domains. More precisely, the aim of this project is to develop the fundamental theory governing optimal control problems, differential games and mean field games in stratified domains and networks, to provide computational methods for their solutions and also to propose a theory of homogenization allowing to pass from microscopic models to macroscopic ones, thereby giving rigorous justifications of the latter. The main objectives include understanding fundamental questions on the structure of optimal trajectories, in particular when moving from one strata to another, the analysis of the value function and its characterization by adequate Hamilton-Jacobi equations, the feedback control, singular perturbations and homogenization. In the particular case of networks, our aim is also to understand the links with conservation laws with discontinuous fluxes. These tools will allow us to tackle a large class of problems in which the dynamics are discontinuous and may depend on the domain where the trajectory takes place. Our project proposes challenging mathematical and numerical studies for optimal control problems, games and mean-field games, and homogenization on stratified structures. Our approaches are based on nonlinear PDEs theory, non-smooth analysis, and advanced numerical methods. Thanks to the expertise of the team members, and inspired by real-life challenging problems, our project will contribute in advancing the theory and will produce open access academic numerical codes. The project is organized in four major methodological axes: optimal control and optimal trajectories, singular perturbation and homogenization, game theory and mean field games and numerical analysis.