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La recherche

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Activités scientifiques du département

Le DMA est à la fois un département d'enseignement et un département de recherche. Cette structuration originale vise notamment à mettre très tôt les élèves au plus près de la recherche en train de se faire.


L'essentielle de publications des membres du département, des thèses et des HDR qui y sont soutenues sont disponibles sur le serveur HAL.

  • 12 July 2024 hal-03144341 publication

    The training of deep residual neural networks (ResNets) with backpropagation has a memory cost that increases linearly with respect to the depth of the network. A simple way to circumvent this issue is to use reversible architectures. In this paper, we propose to change the forward rule of a ResNet by adding a momentum term. The resulting networks, momentum residual neural networks (MomentumNets), are invertible. Unlike previous invertible architectures, they can be used as a drop-in replacement for any existing ResNet block. We show that MomentumNets can be interpreted in the infinitesimal step size regime as second-order ordinary differential equations (ODEs) and exactly characterize how adding momentum progressively increases the representation capabilities of MomentumNets. Our analysis reveals that MomentumNets can learn any linear mapping up to a multiplicative factor, while ResNets cannot. In a learning to optimize setting, where convergence to a fixed point is required, we show theoretically and empirically that our method succeeds while existing invertible architectures fail. We show on CIFAR and ImageNet that MomentumNets have the same accuracy as ResNets, while having a much smaller memory footprint, and show that pre-trained MomentumNets are promising for fine-tuning models.

    Michael E. Sander, Pierre Ablin, Mathieu Blondel, Gabriel Peyré

  • 10 July 2024 hal-04642697 pré-publication

    We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local wellposedness is known since the work of Amann in the 90s, by a semi-group method. We revisit these results in the context of Sobolev spaces modelled on L^2 and exemplify our method with the SKT system, showing the existence of local, non-negative, strong solutions.

    Isabelle Gallagher, Ayman Moussa

  • 30 May 2024 tel-04594602 thèse

    This thesis is devoted to the regularity theory of kinetic Fokker-Planck equations. Firstly, we study the interior regularization effect for the equations with general transport operators and rough coefficients, by revisiting the De Giorgi-Nash-Moser theory and velocity averaging lemmas. The second part addresses the Cauchy problem and the diffusion asymptotics for a kinetic model associated to a nonlinear Fokker-Planck operator. We derive the global well-posedness with instantaneous smoothness effect, and the global diffusion asymptotics quantitively. Finally, we study the existence, uniqueness, and boundary regularization mechanism for the equations in the presence of boundary conditions, including the inflow, diffuse reflection and specular reflection cases.

    Yuzhe Zhu

Les actualités de la recherche

Annonce de conférences, congrès et autres événements scientifiques.


Annales de l’ENS

Les Annales scientifiques de l’École normale supérieure publient 6 fascicules par an. Elles sont éditées par la Société mathématique de France depuis 2008.