Séminaire Equations aux dérivées partielles
organisé par l'équipe Modélisation et contrôle
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Corentin Houpert
Physics-Informed Autoencoders for filling missing values in CO2 measurements
13 janvier 2026 - 14:00Salle de conférences IRMA
Missing values in measurements for carbon dioxide emissions on drained peatlands remains an open challenge for training forecasting techniques to achieve net zero. At the field scale, existing methods struggle to model CO_2 emissions to fill gaps, especially in nighttime measurements. We propose robust Physics-Informed Autoencoders (PIAEs), which combine the generative capabilities of Autoencoders with the reliability of physical models of Net Ecosystem Exchange (NEE) that quantify CO_2 exchanges between the atmosphere and major carbon pools. Our method integrates equations describing the physical processes and associated uncertainties to fill gaps in NEE measurements from eddy covariance (EC) flux towers. In the PIAE, various sensor measurements are encoded into the latent space, and a set of decoders is then used to approximate the ecosystem parameters and the optimal NEE forecast, directed by dynamics described by a stochastic differential equation. These decoders utilize nighttime and daytime NEE models that describe carbon transfer as a Wiener process. Finally, we use a two-phased training routine with two loss functions describing each phase: Mean Squared Error (MSE) and Maximum Mean Discrepancy (MMD) between the measurements and the reconstructed samples. PIAE outperforms the current state-of-the-art Random Forest Robust on the prediction of nighttime NEE measurements on various distribution-based and data-fitting metrics. We present significant improvement in capturing temporal trends in the NEE at daily, weekly, monthly and quarterly scales. -
Raphaël Bulle
Adaptive multi-mesh finite element method for the spectral fractional Laplacian
27 janvier 2026 - 14:00Salle 301
Fractional partial differential equations have gained interest in the last 15 years due to their ability to model non-local behavior (such as e.g. anomalous diffusion) with a relatively small number of parameters. These advantages come with drawbacks from the numerical perspective as these problems raise new challenges regarding simulations. In this talk we are interested in a particular fractional problem, based on the spectral fractional Laplacian operator. More specifically, we look at the discretization and the design of adaptive mesh refinement strategies to efficiently solve such problem numerically. To do so we consider a particular framework based on a rational scheme coupled with a finite element method. We derive an a posteriori error estimator for the finite element discretization error which is then used to steer an adaptive refinement loop. Finally, we will introduce a novel multi-mesh adaptive refinement algorithm taking advantage of the rational scheme to further optimize the discretization. -
Barbara Verfürth
Multiscale methods for wave propagation in spatio-temporal metamaterials
10 février 2026 - 14:00Salle de conférences IRMA
Time modulation of materials has become increasingly popular as a further design approach for metamaterials to induce unusual wave phenomena. In the mathematical modelling, this leads to partial differential equations with coefficients that are highly varying in space and/or time. Since direct numerical simulations are prohibitively expensive, multiscale methods are required to efficiently approximate at least the macroscopic behavior. At the example of the classical wave equation, we will discuss two recent approaches in that direction. One approach is more analytical and aims to deduce higher-order effective equations for temporal multiscale coefficients using asymptotic expansions. The other approach is more numerical, where we present the construction of non-polynomial, multiscale basis functions combined with standard time stepping schemes numerical for spatial multiscale coefficients with additional slow time variations. -
Marie Boussard
Estimation of numerical entropy loss via a projection method
3 mars 2026 - 14:00Salle de conférences IRMA
Ocean circulation models cover a spatial domain up to the planetary scale. Due to computational constraints, numerical simulations therefore rely on relatively coarse meshes, with cell areas of the order of 10 000 km^2. As a consequence, finite volume schemes used for discretization induce a significant numerical entropy loss. This phenomenon leads to unphysical behaviours in the simulation result, such as the mixing of water masses with distinct temperatures and salinities. To address this problem, referred to as "spurious mixing" by oceanographers, localizing and quantifying this numerical entropy loss in high-order codes is crucial. A method was proposed by Aguillon, Audusse, Desveaux and Salomon, but is limited to the one-dimensional case.
I will introduce a method suitable for simulations in two dimensions of space, including high order schemes and in the presence of source terms. This approach relies on projecting a consistent flux onto the set of fluxes satisfying a discrete entropy inequality. The projection is carried out by an optimization algorithm with inequality constraints. Then, I will discuss the theoretical guarantees provided by the method, in particular the establishment of a Lax-Wendroff-type theorem, and conclude with numerical results for the shallow water equations.
This is a joint work with Nina Aguillon and Julien Salomon. -
Guillaume De Romémont
A data-driven learned discretization approach for Finite Volume schemes
10 mars 2026 - 14:00Salle de conférences IRMA
The recent development of Machine Learning (ML) and Deep Learning methods, coupled with recent advances in GPU-based computing defined new promising techniques for the numerical resolution of PDEs entirely solved with ML, as well as tuning existing algorithms for learning corrections or discretizations. In this work, we combine finite volume numerical schemes and neural networks to learn the discretization of the spatial derivatives of partial differential equations (PDEs) in order to better resolve the small spatial scales. We use approximate solutions of the 1D and 2D Euler equations obtained on a finer grid for the reference database in order to learn an optimal spatial discretization on a coarse grid, even with discontinuities in the solution. We post-process the outputs of the neural network to propose an interpretable, entropy consistent subgrid-model with super-resolution capabilities within a second-order finite volume solver without violating physical constraints. -
Corentin Gentil
A linear model of ocean western boundary currents with bathymetry and topography
17 mars 2026 - 14:00Salle de conférences IRMA
The trajectory of ocean western boundary currents is crucial in climate simulations, but the contribution of each physical effect to its path, like wind forcing, stratification, rotation, inertia, geometry of the coast, etc. remains an open question. In this presentation, I will introduce a simplified model for oceanic motion close to the Boussinesq equations, which takes into account two effects that are essential for predicting the trajectory of ocean western boundary currents: stratification and topography. We will see how to construct an approximate solution to this system as a superposition of interior terms on the one hand, and boundary layer terms of different natures on the other hand, due to small parameters. We will study how topography and stratification affect the solution and discuss different asymptotics created by small and large parameters. -
Mabrouk Ben Jaba
À l'optimum, peut-on entendre la ventilation du poumon humain ? Une approche « contrôle optimal ».
24 mars 2026 - 14:00Salle 301
Le poumon constitue une interface d’échange essentielle entre l’air ambiant et le sang, jouant un rôle crucial dans l’oxygénation de ce dernier et l’élimination du dioxyde de carbone. Différentes modélisations mathématiques dans la littérature permettent d’étudier son fonctionnement, certaines faisant intervenir des équations aux dérivées partielles complexes. Une approche alternative consiste à considérer des modèles intégrant l’arbre bronchique dans son ensemble, ce qui constitue le point de vue adopté ici. Notre démarche repose sur l’hypothèse selon laquelle les échanges gazeux sont optimisés pour maximiser l’efficacité du poumon, en accord avec des principes tels que la théorie de l’évolution. Nous cherchons ainsi à retrouver les caractéristiques du cycle respiratoire en formulant un problème d’optimisation (modélisation inverse). Afin d’explorer cette hypothèse et d’évaluer ce principe d'optimalité, nous proposons un modèle basé sur des équations différentielles ordinaires décrivant l’évolution de la concentration de dioxygène dans le poumon et son transport. Dans ce cadre, nous introduisons, analysons et étudions un problème de contrôle optimal visant à caractériser les dynamiques du cycle respiratoire. ——— Travail en commun avec Zakaria BELHACHMI (Univ. Haute-Alsace), Benjamin MAUROY (Univ. Côte d’Azur), Yannick PRIVAT (Univ. Lorraine) et Jean-François SCHEID (Univ. Lorraine).