Institut de recherche mathématique avancée

Résumé : Standard neural network optimization treats all parameters equally, frequently ignoring the inherent separable structure of models that are linear in output weights and nonlinear in hidden weights. This presentation explores Variable Projection (VarPro), a method that explicitly eliminates these linear variables, reducing the task to optimizing only the hidden parameters. Consequently, this reformulates the process into optimizing a subspace on the Grassmannian Manifold, a geometric landscape mathematically proven to be free of spurious local minima.

To resolve practical challenges such as rank deficiency—which occurs when the feature matrix becomes singular during training—a regularized projector manifold is introduced. This regularization ensures the continuous smooth mapping of subspaces while preserving benign critical points. The theoretical framework is validated through the Deep VarPro algorithm, demonstrating robust numerical efficiency in capturing oscillatory behaviors in both low and high-dimensional regressions, as well as successfully solving the heat equation using Physics-Informed Neural Networks (PINNs).

Résumé : Let f, g1, . . . , gd : Rd −→ R be continuous functions. When the functions g1, . . . , gd are of class C1, identifying the d-form f dg1 ∧ · · · ∧ dgd with the continuous function f det(dg) allows one to define the integral ∫_Ω f dg1 ∧ · · · ∧ dgd = ∫_Ω f(x) det(dg(x)) dx, for a bounded Borel set Ω ⊂ Rd. If the functions g1, . . . , gd are not differentiable, it is not clear how to give a meaning to the object f dg1 ∧ · · · ∧ dgd, nor even how to define certain integrals of the form ∫ f dg1 ∧ · · · ∧ dgd. Under regularity assumptions of the type introduced by Züst, we adopt a distributional viewpoint to give a meaning to the object f dg1 ∧ · · · ∧ dgd itself. This approach allows one to define the corresponding integrals, by duality, over more general domains, including sets with fractal boundaries, and to extend integrability results previously obtained by Züst, Alberti–Stepanov–Trevisan, and Bouafia. This talk is based on the preprint available at https://arxiv.org/abs/2510.20427.

Résumé : Les connexions du type Painlevé V sont une classe particulier de connexions irrégulières de rang deux sur la sphère de Riemann, avec un précise diviseur des poles. Leur espace des modules est doté d'un feuilletage en courbes dont les feuilles contiennent les connections avec même monodromie. Ce feuilletage est induit par une équation différentielle classique, l'équation Painlevé V. Le but de cet exposé est de présenter une compactification de l'espace des modules qui nous permettra d'étudier le comportement asymptotique des feuilles isomonodromiques sur les composants du bord.

Résumé : Continuing from last year, on my side quest into the land of cobordisms, I come to the question of why we study cobordisms? Interestingly, cobordisms are closely related to maps between spheres. In this talk I will speak about what are homotopy and homology groups and how they are really just two definitions of holes. I will speak about the homotopy groups of spheres and how studying them we may come to framed cobordisms. Finally, should there be time I will explain how the homotopy of spheres and cobordisms may motivate more advanced homotopy topics such as stable homotopy and rational homotopy.

Résumé : In this presentation, I will introduce a hybrid finite volume method on general polygonal and polyhedral meshes for the modeling of an anisotropic cross-diffusion system arising from a mesoscopic stochastic process describing diffusion in solids, under a size-exclusion constraint.
This system possesses an entropy structure, which is exploited to define the numerical scheme in terms of (discrete) entropy variables, and is thus preserved at the discrete level.
This structure makes it possible to prove the existence of nonnegative discrete solutions satisfying the size-exclusion constraint, as well as mass conservation, and to establish the convergence of the scheme under mesh refinement.
To the best of our knowledge, this is the first work proposing and analyzing a structure-preserving hybrid finite volume scheme for anisotropic cross-diffusion systems on general polygonal and polyhedral meshes.

The preprint associated with this presentation is:
V. Ehrlacher, A. Massimini, J. Moatti. Structure-preserving hybrid finite volume scheme for an anisotropic cross-diffusion system, 2026. Preprint, HAL : hal-05589824