This talk illustrates the application of stochastic stability methods - such as couplings,
optimal transport, irreducibility, and Lyapunov techniques - in establishing a rigorous
mathematical framework for the ergodicity of affine processes in the modelling of stochastic
interest rates, default intensities, and stochastic volatility. Such ergodicity results are
shown to be crucial for the estimation of parameters in the mean-reversion regime of affine
processes.

Firstly, I will explain my thesis topic and the motivations driving us to study this problem. Secondly, I will describe how we approached this problem.

Dans cette présentation nous discuterons des réseaux de neurones informés par la physique (PINNs) qui sont actuellement au cœur de recherches nourries mêlant les modélisations mécanistes (EDO / EDP) à l’apprentissage machine et aux statistiques. Nous commencerons par une présentation des concepts fondamentaux des PINNs : a priori physique, dérivation automatique, et réseaux de neurones. Nous nous intéresserons ensuite à deux pistes de recherche actuellement suivies au sein de MIA Paris-Saclay.

La distance de Wasserstein est un outil puissant pour comparer des distributions de probabilité et est largement utilisée pour la classification et la récupération de documents dans les tâches de NLP (Natural Language Processing). En particulier, elle est connue sous le nom de Word Mover’s Distance (WMD) dans la communauté NLP.

The talk will review recent progress on turbulent transport based on Ito-Stratonovich calculus. The general principle is a new way to reformulate turbulent transport as an additional diffusion, by taking a suitable scaling limit based on stochastic calculus. The idea has been applied with success to several problems, like the transport in passive scalars, transport of passive vector fields (like the magnetic field), the effect of turbulence on coalescence of droplets, the effect of turbulent small scales on the large scales of the fluid itself.

After reviewing the principles of so-called sampling methods for imaging the geometry of an inclusion from measurements of scattered waves, we will highlight the role of some specific frequencies where some transparent incident waves may exist. While these frequencies must be avoided for these imaging algorithms, their link to the material properties can be a key property for looking inside highly cluttered media.

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprises, among the others, equations involving both standard and Caputo time-derivative, and diffusion operators as the classic and fractional Laplacian, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.

Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems, whose basic concept is to embed physical laws to constrain/inform neural networks, with the need of less data for training a reliable model. In this presentation, we introduce physics-informed neural networks (PINNs) and we explore the resolution of the Stefan problem using this method.