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Flow Matching is a recent framework for learning continuous transformations between probability measures. The method constructs a time-dependent velocity field whose flow transports a source distribution to a target distribution, and whose training reduces to a simple regression problem on paired samples. This simulation-free objective makes Flow Matching an attractive alternative to continuous normalizing flows and diffusion models. This tutorial provides a self-contained and mathematically rigorous introduction to Flow Matching, aimed at applied mathematicians. Starting from the continuity equation, we establish the theoretical foundations linking velocity fields, probability paths, and flows, and explain how Flow Matching arises from a particular construction based on couplings of probability measures. We carefully state the assumptions under which the induced ordinary differential equation defines a unique flow and yields a valid pushforward between distributions, and we illustrate the limitations of the theory through explicit counterexamples. We derive closed-form velocity fields in several important settings, including one-dimensional distributions, Gaussian and Gaussian mixture models, and semi-discrete targets, and we clarify the connections with score matching, diffusion models, and optimal transport. Throughout the paper, theoretical results are complemented by reproducible numerical experiments designed to build intuition and illustrate practical behavior. Our goal is to provide readers with both a solid mathematical understanding of Flow Matching and concrete tools for its application.

The work presented here is concerned with breaking water waves, a well-known phenomenon arising as an oceanic wave approaches the shore: its crest starts to move faster than the trough up front, which ultimately leads to the appearance of an overhanging region that quickly curls over while falling down until it collides with the water lying below. An important contemporary issue concerns the incorporation of the viscous dissipation associated with the breaking into the many models that have been introduced to describe the ocean. This is mostly done empirically. In the present work, we follow a different path: we aim at modelling wave breaking up to the free surface self-intersection (the splash singularity), relying thus on a more geometrical approach to the subject. The first part of this thesis will be devoted to the motivation of a set of equations that describes overhanging waves in the inviscid irrotational regime, with either a one-dimensional or a two-dimensional free surface. This is done by setting aside the commonly used Eulerian framework and working in (pseudo)Lagrangian coordinates instead. This should be seen as an extension of the Zakharov-Craig-Sulem formulation of the Water Waves problem. The non-canonical Hamiltonian structure of these partial differential equations is investigated and it is shown that in the absence of breaking, they can be reduced to the usual set of equations. Emphasis is put on the various physical assumptions that are made along the way. In a second moment, we come back to these very hypotheses and put them to the test. This is done numerically using a Navier-Stokes based computational framework based on the Finite-Element Method (FEM). The major novelty compared to other studies lies in the use of the Arbitrary Lagrangian-Eulerian method (ALE), which diminishes the interpolation error greatly. The viscosity can therefore be decreased to values that allow the comparison with the inviscid solution (computed using another method, based on potential theory in the complex plane) to be carried out. Over a flat topography, it is found that both the free-surface and bed boundary layers are sufficiently well-behaved as to not perturb the bulk irrotational flow. Water being characterised by a relatively small viscosity, the consequence is that, in this regime the inviscid models accurately describe the oceanic flow. We do not prove this assertion rigorously, however. Difficulties seem to arise, however, when a non-flat topography is considered. Indeed, the typical velocities associated with the wave are high enough to eventually trigger boundary layer separation near curved-enough portions of the bed, resulting in vorticity being shed in the initially irrotational flow, far from the topography. The convergence to the inviscid solution is therefore compromised.