• Deep learning-based reduced order models for parametrized PDEs

    — Stefania Fresca

    31 janvier 2023 - 14:00Salle de conférences IRMA

    The solution of differential problems by means of full order models (FOMs), such as, e.g., the finite element method, entails prohibitive computational costs when it comes to real-time simulations and multi-query routines. The purpose of reduced order modeling is to replace FOMs with reduced order models (ROMs) characterized by much lower complexity but still able to express the physical features of the system under investigation. Conventional ROMs anchored to the assumption of modal linear superimposition, such as proper orthogonal decomposition (POD), may reveal inefficient when dealing with nonlinear time-dependent parametrized PDEs, especially for problems featuring coherent structures propagating over time. To overcome these difficulties, we propose an alternative approach based on deep learning (DL) algorithms, where tools such as convolutional neural networks (CNNs) are used to build an efficient nonlinear surrogate. In the resulting DL-ROM, both the nonlinear trial manifold and the nonlinear reduced dynamics are learned in a non-intrusive way by relying on DL models trained on a set of FOM snapshots, obtained for different parameter values [Fresca et al. (2021a), Fresca et al. (2022)]. Accuracy and efficiency of the DL-ROM technique are assessed in several applications, ranging from cardiac electrophysiology [Fresca et al. (2021b)] to fluid dynamics [Fresca et al. (2021c)], showing that new queries to the DL-ROM can be computed in real-time. Finally, with the aim of moving towards a rigorous justification on the mathematical foundations of DL-ROMs, error bounds are derived for the approximation of nonlinear operators by means of CNNs. The resulting error estimates provide a clear interpretation on the role played by the hyperparameters of dense and convolutional layers. Indeed, by exploiting some recent advances in Approximation Theory, and unvealing the intimate relation between CNNs and the discrete Fourier transform, we are able to characterize the complexity of the neural network in terms of depth, kernel size, stride, and number of input-output channels [Franco et al. (2023)]. References S. Fresca, A. Manzoni, L. Dede’ 2021a A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. Journal of Scientific Computing, 87(2):1-36. S. Fresca, A. Manzoni 2022 POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Computer Methods in Applied Mechanics and Engineering, 388, 114181. S. Fresca, A. Manzoni L. Dede’, A. Quarteroni 2021b POD-enhanced deep learning-based reduced order models for the real-time simulation of cardiac electrophysiology in the left atrium. Frontiers in Physiology, 12, 1431. S. Fresca, A. Manzoni 2021c Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models. Fluids, 6(7), 259. N. R. Franco, S. Fresca, A. Manzoni, P. Zunino 2023 Approximation bounds for convolutional neural networks in operator learning. Neural Networks, Accepted.
  • TBA

    — Andrea Thomann

    7 février 2023 - 14:00Salle de séminaires IRMA

  • TBA

    — Angèle Niclas

    28 février 2023 - 14:00Salle de séminaires IRMA

  • Demi-journée de l'équipe MoCo

    7 mars 2023 - 14:00A confirmer

  • Analysis of the Shallow Water equations with two velocities

    — Nelly Boulos Al Makary

    14 mars 2023 - 14:00Salle de conférences IRMA

  • TBA

    — Olivier Hurisse

    28 mars 2023 - 14:00Salle de conférences IRMA

  • Maximisation des valeurs propres du Laplacien avec condition de Neumann

    — Eloi Martinet

    4 avril 2023 - 14:00Salle de conférences IRMA

    On s'intéresse au problème d'optimisation de formes consistant à maximiser les valeurs propres du Laplacien avec conditions de Neumann homogènes. Ces valeurs propres interviennent notamment dans des problèmes acoustiques ou thermiques et sont en particulier liées à la "hot spot conjecture". Contrairement aux valeurs propres de Dirichlet, celles associées au problème de Neumann sont de nature plutôt instables, ce qui rend le problème d'optimisation difficile. On verra comment certaines explorations numériques du problème pour des domaines du plan et de la sphère ont permis de mettre en évidence certaines propriétés des optima.