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.