Publications

2026

• Spatio-temporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures
  
  • Zanaglia Lucas
  • Garnier Josselin
  • Michel Claire
  • Doya Valérie
  • Ferraro Mario
  • Wabnitz Stefan
  • Carusotto Iacopo
  • Picozzi Antonio
  Physical Review A, American Physical Society, 2026, 113, pp.063533. Optical thermalization has been recently studied theoretically and experimentally in the 2D spatial evolution of (quasi-)monochromatic light waves propagating in multimode fibers. In this work, we investigate the spatio-temporal equilibrium properties of incoherent multimode optical waves through the analysis of the (2+1)D Bose-Einstein thermal distribution and the corresponding classical Rayleigh-Jeans approximation. In the classical regime, we perform numerical simulations of the nonlinear Schr\"odinger equation and demonstrate relaxation toward the spatio-temporal Rayleigh–Jeans equilibrium state, as described by the corresponding wave turbulence kinetic equation. Remarkable adiabatic cooling phenomena stemming from the high-frequency tails of the Rayleigh-Jeans distribution are discussed and the consequent limitations of the classical approximation are highlighted. To overcome these issues and properly include quantum effects, we make use of a quantum version of the nonlinear Schr\"odinger equation that is obtained from the general quantum theory of light propagating in nonlinear waveguides. The associated kinetic equation describes relaxation toward the spatio-temporal Bose-Einstein equilibrium distribution with a fully regular UV behaviour. The analysis of thermodynamic equilibrium properties reveals a strong dependence on the specific dispersion regime under consideration. In the anomalous dispersion regime, the system relaxes to positive-temperatures equilibrium states: as the number of modes of the waveguide increases, the fundamental spatial mode becomes macroscopically populated, while its temporal spectrum undergoes significant narrowing, ultimately leading to complete (2+1)D spatio-temporal condensation in the thermodynamic limit. By contrast, in the normal dispersion regime the system evolves toward negative-temperature equilibrium states characterized by a hybrid structure: the spatial equilibrium displays an inverted modal population, whereas the temporal spectrum remains peaked around the fundamental (carrier) optical frequency. In this regime, we predict that spatio-temporal light waves exhibit a phase transition to Bose-Einstein condensation at negative temperatures, which occurs by increasing the temperature above a negative critical value. Our work opens new avenues for future research, including the possibility for a dual spatio-temporal beam cleaning through full spatio-temporal light condensation, and lay the groundwork for the development of spatio-temporal optical thermodynamics. (10.1103/bf7m-chkd)
  DOI : 10.1103/bf7m-chkd
• ExceedGAN: Simulation above extreme thresholds using Generative Adversarial Networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  Extremes, Springer Verlag (Germany), 2026. This paper devises a novel neural-inspired approach for simulating multivariate extremes. Specifically, we propose a GAN-based generative model for sampling multivariate data exceeding large thresholds, giving rise to what we refer to as the ExceedGAN algorithm. Our approach is based on approximating marginal log-quantile functions using feedforward neural networks with eLU activation functions specifically introduced for bias correction. An error bound is provided {on the margins}, assuming a $J$th order condition from extreme value theory. The numerical experiments illustrate that ExceedGAN outperforms competitors, both on synthetic and real-world data sets. The code is available at \url{https://github.com/michael-allouche/extreme-value-GAN} (10.1007/s10687-026-00528-9)
  DOI : 10.1007/s10687-026-00528-9
• Numerical Analysis of differential equations on weighted Sobolev spaces: beyond classical orthogonal polynomials
  
  • Breden Maxime
  • Chu Hugo
  , 2026. We lay mathematical foundations for the Numerical Analysis of differential equations on Sobolev spaces weighted by a Gibbs probability measure $\nu$(dx) = e^-V(x)dx/$\mathcal{Z}$ on the real line. Over recent decades, the Functional Analysis of these spaces has been thoroughly developed to study Schrödinger-type equations and diffusion processes. While such equations should therefore be amenable to a numerical resolution with respect to orthogonal polynomials, this feat has only ever been achieved with respect to classical bases. We bridge this gap by showing that such equations can be solved with respect to suitable bases by factorising their leading linear component. In particular, we propose a new natural notion of Sobolev orthogonal polynomials, simpler and more tractable than those arising from the usual Sobolev inner product. In the case of $V$ being an even polynomial, we further establish quantitative estimates for the compactness of the embedding $H^1(\nu)\hookrightarrow L^2(\nu)$, uncovering a connection with the growth of the Jacobi recurrence coefficients, which are solutions of corresponding Painlevé-type discrete equations. As an application, we rigorously and tightly enclose solutions of the Gross--Pitaevskii equation with sextic potential and rigorously demonstrate the phenomenon of stochastic resonance via a computer-assisted proof.
• SAM-Guide : vers un premier Laser Run à destination de sportifs non-voyants
  
  • Romeo-Pakker Katerin
  • Huet Sylvain
  • Faugloire Elise
  • François Alouges
  • Abdellatif Ennaji
  • Christèle Lecomte
  • Wenqi Luo
  • Pellerin Denis
  • Fons Coline
  • Bruno Mantel
  • Mathieu Thomas
  • Sylvain Ferrand
  • Clément Francqueville
  • Graff Christian
  , 2026. Le projet ANR SAM-Guide vise à rendre le Laser Run praticable par des personnes aveugles et malvoyantes grâce à la combinaison multimodale de systèmes d'aide à la navigation et à l'atteinte de cible. Le dispositif fournit un retour auditif et/ou tactile en temps réel sur la position du corps et l'orientation du pistolet vers une cible virtuelle. Nous décrivons l'architecture du système et rapportons les premiers résultats obtenus auprès de participants avec ou sans capacité visuelle.
• Counterfactually Fair Regression via Optimal Transport
  
  • Lince Marie Generali
  • Gaucher Solenne
  • Vie Jill-Jênn
  • Loiseau Patrick
  , 2026. We consider the problem of learning a counterfactually fair regressor. We adopt a causal uncertainty view in which counterfactual fairness is defined with resampled noise. We focus on obtaining theoretical fairness guarantees for a new post-processing estimator. We begin by showing that counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This allows us to provide a closed-form expression of the optimal fair regressor via a barycentric quantile map. In order to handle continuous latent variables, we propose a discretized post-processing method. Then, under mild regularity assumptions, we prove high-probability finite-sample fairness guarantees for our estimator, providing an unfairness decay at rate Õ(n -1/3 ), and establishing a matching risk bound of order Õ(n -1/3 ). We provide a matching lower bound on the excess risk of almost fair predictions. Finally, we extend our results to the setting of relaxed counterfactual fairness. We validate our approach on real-world and synthetic data.
• Finite element modelling for the reproduction of dynamic OCE measurements in the cornea
  
  • Merlini Giulia
  • Imperiale Sébastien
  • Allain Jean-Marc
  Journal of the Mechanics and Physics of Solids, Elsevier, 2026, 206, pp.106363. Recent advances in dynamic elastography, particularly through optical coherence tomography combined with transient excitations have enabled rapid, localized, and non-invasive mechanical data acquisition of the cornea. This dataopens the path to early-detection of pathologies and more accurate treatment. However, the analysis of the wave propagation is a complex mechanical problem: the cornea is a structure under pressure, with non-linear material behavior. Thus, computational analysis are needed to extract mechanical parameters from the data. In this study, we present a time-dependent finite element model for the reproduction of transient shear wave elastographic measurements in the cornea. The mechanical problem consists in a smallamplitude wave propagating in the cornea, largely deformed by intraocular pressure in physiological conditions. The model accounts for anisotropic, hyperelastic, and incompressible behavior of the cornea, as well as its accurate geometry, and the preloaded condition. We have implemented two different numerical approaches to solve first the static non-linear inflation of the cornea and then the linear wave propagation problem to reproduce the measurements. We investigate the impact of material anisotropy and prestress on wave propagation and demonstrate that intraocular pressure critically influences shear wave velocity. Additionally, by introducing a localized mechanical defect to simulate a pathological defect, we show that simulated shear wave can detect and quantify mechanical weaknesses, suggesting potential as a diagnostic tool to assess corneal health. (10.1016/j.jmps.2025.106363)
  DOI : 10.1016/j.jmps.2025.106363
• Validated Enclosure of Renormalization Fixed Points Via Chebyshev Series and the DFT
  
  • Breden Maxime
  • Gonzalez Jorge
  • Mireles James Jason D.
  Foundations of Computational Mathematics, Springer Verlag, 2026. This work develops a computational framework for proving existence, uniqueness, isolation, and stability results for degree d, real analytic, unimodal functions fixed by m-th order Feigenbaum-Coullet-Tresser-Cvitanović renormalization operators using Chebyshev series approximation. Here the order m of the operator refers to the number of function compositions involved in its definition, and the degree d of the fixed function is the number of derivatives vanishing at its (unique) critical point. The advantage of Chebyshev series is that they are naturally adapted to spaces of real analytic functions, in the sense that they converge on ellipses containing real intervals rather than on disks in C. This facilitates the development of a functional analytic approach independent of order and degree. The main disadvantage of working with Chebyshev series is that the operations of rescaling and composition, essential to the definition of the renormalization operators, are less straightforward for Chebyshev than for Taylor series. These difficulties are overcome via a combination of a-priori information about decay rates in Banach spaces of rapidly decaying coefficient sequences, with a-posteriori estimates on Chebyshev interpolation errors for analytic functions. We illustrate the utility of the method by proving existence, local uniquness, global non-uniqueness results, and interval enclosures of universal constants for fixed points of varying orders and degrees. (10.1007/s10208-026-09763-9)
  DOI : 10.1007/s10208-026-09763-9
• Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method
  
  • Du Kai
  • Ren Zhenjie
  • Suciu Florin
  • Wang Songbo
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2026, 205, pp.103782. For a certain class of McKean-Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean-Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron. (10.1016/j.matpur.2025.103782)
  DOI : 10.1016/j.matpur.2025.103782
• Hybrid FEM/IPDG semi-implicit schemes for time domain electromagnetic wave propagation in non cylindrical coaxial cables
  
  • Beni Hamad Akram
  • Imperiale Sébastien
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2026. In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.<p>Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.</p>
• Modelling of relative velocity, velocity fluctuations and their interactions for two-fluid models by Stationary Action Principle
  
  • Haegeman Ward
  • Orlando Giuseppe
  • Kokh Samuel
  • Massot Marc
  ESAIM: Proceedings and Surveys, EDP Sciences, 2026. The objective of this contribution is the derivation of a two-fluid model including a relative velocity between the two phases and velocity fluctuations, describing pseudo-turbulent effects, as internal variables based on Stationary Action Principle. The variational derivation, used to obtain the model, relies on the variation of a single trajectory related to the mass-weighted average velocity under the barotropic assumption. The model is hyperbolic, satisfies a second principle of thermodynamics, and admits either linearly degenerate or genuinely nonlinear characteristic fields. Moreover, the variational approach yields a fully closed model and its non-conservative products are uniquely defined for weak solutions in 1D, i.e. jump conditions can be derived. In the laminar case, when velocity fluctuations are negligible, we recover previously derived multi-fluid models which have been analyzed in several contributions. As such, the present framework allows for an original extension of the existing models to include velocity fluctuations of each phase for pseudo-turbulent flows, their coupling with the relative velocity between phases, as well as dissipative effects compatible with the thermodynamics of irreversible processes. Eventually, we provide a discussion of the limitations of the proposed model, especially regarding the extension to the open problem of non-barotropic flows.
• Stability analysis of a new curl-based full field reconstruction method in 2D isotropic nearly-incompressible elasticity
  
  • Chibli Nagham
  • Genet Martin
  • Imperiale Sébastien
  Inverse Problems, IOP Publishing, 2026. In time-harmonic elastography, the shear modulus is typically inferred from full field displacement data by solving an inverse problem based on the time-harmonic elastodynamic equation. In this paper, we focus on nearly incompressible media, which pose robustness challenges, especially in the presence of noisy data. Restricting ourselves to 2D and considering an isotropic, linearly deforming medium, we reformulate the problem as a non-autonomous hyperbolic system and, through theoretical analysis, establish existence, uniqueness, and stability of the inverse problem. To ensure robustness with noisy data, we propose a least-squares approach with regularization. The convergence properties of the method are verified numerically using in silico data.
• Nonlinear model calibration through bifurcation curves
  
  • Mélot Adrien
  • Denimal Goy Enora
  • Renson Ludovic
  Mechanical Systems and Signal Processing, Elsevier, 2026, 242, pp.113589. Nonlinear systems exhibit a plethora of complex dynamic behaviours that are difficult to model and predict accurately. This difficulty often arises from a lack of knowledge of the physics that induces the nonlinear behaviours and the strong sensitivity of the nonlinear dynamics to parameter variation. We introduce in this paper a methodology to carry out nonlinear model updating based on bifurcations. The proposed approach involves minimising the distance between experimental and numerical bifurcation curves, which are key dynamic features that define stability boundaries and regions of multi-stability. For the model, bifurcation curves are computed via standard numerical bifurcation tracking analyses. In the experiment, we use control-based continuation to obtain the data. The approach is first demonstrated on a Duffing and a beam system using synthetic data, before being applied to experimental data collected on a base-excited energy harvester with magnetic nonlinearity.
• Improved well-posedness for the limit flow of differentiation of roots of polynomials
  
  • Bertucci Charles
  • Pesce Valentin
  , 2026. In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberg [32] on the real line and which represents the evolution of the density of the roots of polynomials under differentiation. After integrating the partial differential equation in question, we observe that it can be treated with the theory of viscosity solutions. This equation at hand is a non linear parabolic integro-differential equation which involves the elliptic operator called the half-Laplacian. Due to the singularity of the equation, we restrict our study to strictly positive initial condition. We obtain a comparison principle for solutions of the primitive equation which yields uniqueness, existence, continuity with respect to initial condition. We also present heuristics to justify that the system of particles indeed approximates the solution of the equation.
• A stochastic use of the Kurdyka-Lojasiewicz property: Investigation of optimization algorithms behaviours in a non-convex differentiable framework
  
  • Fest Jean-Baptiste
  • Repetti Audrey
  • Chouzenoux Emilie
  Foundations of Data Science, American Institute of Mathematical Sciences, 2026, 9, pp.164-191. Asymptotic analysis of generic stochastic algorithms often relies on descent conditions. In a convex setting, some technical shortcuts can be considered to establish asymptotic convergence guarantees of the associated scheme. However, in a non-convex setting, obtaining similar guarantees is usually more complicated, and relies on the use of the Kurdyka-Łojasiewicz (KŁ) property. While this tool has become popular in the field of deterministic optimization, it is much less widespread in the stochastic context and the few works making use of it are essentially based on trajectory-by-trajectory approaches. In this paper, we propose a new framework for using the KŁ property in a non-convex stochastic setting based on conditioning theory. We show that this framework allows for deeper asymptotic investigations on stochastic schemes verifying some generic descent conditions. We further show that our methodology can be used to prove convergence of generic stochastic gradient descent (SGD) schemes, and unifies conditions investigated in multiple articles of the literature. (10.3934/fods.2025016)
  DOI : 10.3934/fods.2025016
• Spatio-temporal thermalization and adiabatic cooling of guided light waves
  
  • Zanaglia Lucas
  • Garnier Josselin
  • Carusotto Iacopo
  • Doya Valérie
  • Michel Claire
  • Picozzi Antonio
  Physical Review Letters, American Physical Society, 2026, 136, pp.053802. We propose and theoretically characterize three-dimensional spatio-temporal thermalization of a continuous-wave classical light beam propagating along a multi-mode optical waveguide. By combining a non-equilibrium kinetic approach based on the wave turbulence theory and numerical simulations of the field equations, we anticipate that thermalizing scattering events are dramatically accelerated by the combination of strong transverse confinement with the continuous nature of the temporal degrees of freedom. In connection with the blackbody catastrophe, the thermalization of the classical field in the continuous temporal direction provides a novel intrinsic mechanism for adiabatic cooling and spatial beam condensation. This process of adiabatic cooling is distinct from other mechanisms of thermalization and provides new insights into the dynamics of far-from-equilibrium closed systems and their route to thermalization. (10.1103/mqzh-w2gh)
  DOI : 10.1103/mqzh-w2gh
• Nonparametric regression on Riemannian manifolds under an alpha -mixing process
  
  • Wiem Nefzi
  • Salah Khardani
  • Françoise Yao Anne
  Communications in Statistics - Theory and Methods, Taylor & Francis, 2026, pp.1-16. This paper investigates the asymptotic properties of a kernel estimator of the regression function linking a real-valued response Y to a covariate X taking values in a Riemannian submanifold M. The estimator is an adaptation of the nonparametric regression estimator introduced by Pelletier (2006) for i.i.d. data on manifolds to the case where the observations (X t , Y t ) t∈Z form a stationary α-mixing process. Under suitable geometric regularity and mixing assumptions, we derive explicit expressions for the bias and the variance of the estimator, obtain the optimal bandwidth that minimizes the mean squared error, and establish the optimal rate of convergence in mean squared error. As a consequence, we prove both pointwise and uniform convergence in probability of the estimator on compact Riemannian manifolds under dependence. (10.1080/03610926.2026.2618625)
  DOI : 10.1080/03610926.2026.2618625
• Quantitative sensitivity analysis for Fokker-Planck equation with respect to the Wasserstein distance
  
  • Morange Martin
  , 2026. We analyze the sensitivity of solutions to the Fokker-Planck equation with respect to some unknown parameter. Our main result is to provide quantitative upper bounds for the $p$-Wasserstein distance $\mathcal{W}_p$ between two solutions with different parameters, for every $p \geq 2$. We are able to give two proofs of this result, the first relying on synchronous coupling between two solutions of an SDE, and another one that relies on the differentiation of Kantorovitch dual formulation of optimal transport. We also provide more specific bounds in the case of the overdamped Langevin process, for which we are able to compare convergence to the invariant measure and sensitivity to the parameter.