Studying the global dynamics of nonlinear differential equations is a very challenging task. Even proving the existence of non trivial equilibria or periodic orbits is often difficult, at least if one moves away from perturbative or asymptotic parameter regimes, to say nothing of establishing connecting orbits between these types of solutions, nor of proving the existence of more complex, e.g. chaotic, dynamics. In this context, there is often a large gap between what can be observed and conjectured from numerical simulations, and what can actually be proven mathematically. The goal of this project is to bridge this gap by developing new computer-assisted techniques, essentially allowing to turn numerical simulations into mathematical theorems. Computer-assisted proofs have recently risen in popularity in the fields of differential equations an dynamical systems, and can now be used to study ODEs in a rather systematic way. Our main objective is to develop such tools for studying some classes of PDEs and SDEs, allowing us to obtain new results that are currently out of reach of purely analytical techniques. Here are the principal directions we want to investigate: - Existence and precise description of patterns (i.e. non homogeneous steady states) in quasilinear PDEs; - Stability properties, of the above-mentioned patterns, but more generally of solutions obtained using computer-assisted techniques; - Large deviation estimates for finite time Lyapunov exponents of SDEs, allowing to predict a sign change of the first Lyapunov exponent; - A unified framework to describe computer-assisted proofs based on finite elements methods, finite volume methods and on spectral techniques. We plan to focus on some specific problems, like chemotaxis models in population dynamics, but the new computer-assisted techniques produced by this project will by nature be rather robust, and will therefore provide a new set of tools for studying a wide array of nonlinear differential equations.

The goal of this project is to study the most probable trajectory followed by an interacting particle system when a spontaneous fluctuation is observed. In its most basic form, this is known as the Schrödinger problem. In their general form, the problems at the heart of this project are formulated by means of large deviations theory. Notable examples we shall study include classical models in statistical mechanics and systems of interacting diffusion processes. We have two main objectives. The first is to study the infinite dimensional HJB equations associated to a general Schrödinger problem and to carry out a rigorous analysis of the optimality conditions both in the form of coupled PDE systems and in terms of forward backward pathwise stochastic equations. The second goal is to obtain entropy dissipation estimates along Schrödinger bridges to understand their ergodic behaviour in connection with the obtention of a novel class of functional inequalities and the turnpike property.

RheoSuNN project brings together researchers in the domains of mathematics, numerical analysis, HPC and mechanics, interested in numerical simulations of dense suspensions of rigid particles embedded in a Stokes fluid. The scientific program of this project is based on the following objective: achieve a numerical rheological study in order to understand the interphase stress contribution to the total fluid stress. The last year of the project is dedicated to this task. It consists in a challenging study, for which no result is available (neither experimental, nor theoretical, nor numerical). The key point to make a breakthrough in the understanding of the interphase stress is to compute the velocity and pressure fields in the whole fluid domain for a dense suspension, to model carefully the multi-body lubrication and contact interactions and their feedback on the flow. From a computational point of view, it leads to suspension simulations with density around 50%, containing up to about one million of suspended particles. The corresponding mesh size for the direct fluid solver is around 500 x 2 000 x 10 000. Despite the wide range of available techniques, there is a great need of designing new mathematical models and numerical methods to achieve this numerical study. The project is based on two existing codes developed by members of the project: a fluid/particle solver (CAFES) and a contact solver (SCoPI). To reach the target, the project first two years are dedicated to the two codes coupling, together with their optimisation and the design of new numerical methods to be able to take into account stiff close interactions and their effects on the whole flow, for any shape of particles. This leads to highly coupled fluid/particle problems to be solved in an implicit way, in presence of singularities. This problem need fine numerical expertise to be treated in an efficient way. We plan to deliver a code including the implicit methods developed in the project. In order to run simulations for real physical configurations, it is crucial to develop a numerical code which is optimised and able to run with good scalability on the national and European HPC infrastructures proposed by GENCI and PRACE. This code will allow to reach more precise and dense simulations than the existing ones and will enable the study of open questions about the rheology of suspension. The code we develop allows to deal with non-spherical particles (more precisely with superellipsoids) while taking lubrication into account with retroaction on the flow, which is not possible with the other existing codes. As a consequence, even though we focus in this project on the interphase stress, it opens the opportunity for numerous other interesting and original numerical rheological studies of non-spherical particles suspensions (like rods, fibres or micro-swimmers). Moreover, taking in a very fine way lubrication phenomenon into account, it can be a reference code, providing precise results to study new rheological phenomenon or to compute reference solutions to be compared to other methods. It can also be a first step towards direct numerical simulation of suspensions of particles embedded in non-Newtonian fluids or of active suspensions which are two very active domains of research.

Until recently, most of the major advances in machine learning and decision making have focused on a centralized paradigm in which data are aggregated at a central location to train models and/or decide on actions. This paradigm faces serious flaws in many real world cases. In particular, centralized learning risks exposing user privacy, makes inefficient use of communication resources, creates data processing bottlenecks, and may lead to concentration of economic and political power. It thus appears most timely to develop the theory and practice of a new form of machine learning that targets heterogeneous, massively decentralized networks, involving self-interested agents who expect to receive value (or rewards, incentive) for their participation in data exchanges. OCEAN will develop statistical and algorithmic foundations for systems involving multiple incentive-driven learning and decision making agents, including uncertainty quantification at the agent's level. OCEAN will study the interaction of learning with market constraints (scarcity, fairness), connecting adaptive microeconomics and market-aware machine learning. OCEAN builds on a decade of joint advances in stochastic optimization, probabilistic machine learning, statistical inference, Bayesian assessment of uncertainty, computation, game theory, and information science, with PIs having complementary and internationally recognized skills in these domains. OCEAN will shed new light on the value and handling of data in a competitive, potentially antagonistic multi-agent environment, and develop new theories and methods to address these pressing challenges. OCEAN requires a fundamental departure from standard approaches and leads to major scientific interdisciplinary endeavors that will transform statistical learning in the long term while opening up exciting and novel areas of research.

Domain Adaptation (DA) is a fundamental problem in statistics, Machine Learning (ML), and data science where one wants to estimate a predictive model from labeled training data in the presence of a shift or change in the properties of the testing data. This problem is very common in practical applications and challenging due to the lack of labels in the shifted test data. Despite an active research community, DA methods remain rarely used in practice. The objective of project MATTER is to tackle theoretical and practical bottlenecks that prevent the wider use of DA in ML applications. MATTER focuses on the root of the DA problem: the estimation of the shift between domains, which will be achieved using optimal transport, manifold estimation, and physical modeling. This will lead to novel and interpretable shift classification and estimation methods for DA. These results will be used to implement robust validation procedures that are still lacking in the DA community and pave the way for the first Automatic DA framework. MATTER will also address the general DA problem where multiple shifts are present between multiple datasets and propose interpretable and scalable (distributed) methods. MATTER will finally investigate the problem of Heterogeneous DA that can occur between heterogeneous datasets across devices and between structured data such as graphs. The proposed methods will be validated with a new open benchmark framework on several types of data (computer vision, biomedical, audio). They will also be evaluated on a flagship biomedical application, sleep stage classification, where adaptation to the specificities of the subjects is necessary. A major output of project MATTER will be an open-source toolbox containing implementations of DA methods, the benchmark, and the first-ever AutoDA software. By addressing both theoretical and more pragmatic aspects of the field, project MATTER is a unique opportunity to unlock the potential of DA.