Publications

2026

• Méthode de méta-modélisation par décomposition de domaines et processus Gaussiens pour la prédiction de fonctions de réponse en fréquences de structures incertaines
  
  • Denimal Goy Enora
  , 2026. L’utilisation de méthodes de méta-modélisation est une stratégie classique pour limiter le coût numérique associé aux études de propagation d’incertitudes. Une quantité d’intérêt classique pour les systèmes dynamiques est la Fonction de Réponse en Fréquence (FRF). Ce travail présente une stratégie de méta-modélisation basée sur de la décomposition de domaines, en exploitant les phénomènes de résonance et anti-résonance, et des processus Gaussiens pour pouvoir construire efficacement un méta- modèle de FRF. Les résultats sont illustrés sur un modèle éléments finis de poutre.
• Development of a 2D test bench for characterising the dynamic mechanical response of a solid rocket propellant under rapid depressurisation
  
  • Distelzwey Léo
  • Levard Quentin
  • François Laurent
  • Voreux Olivier
  , 2026. The ignition of a solid rocket propellant is a critical moment during which the material is subjected to a rapid and large variation in chamber pressure. This occurs especially when the operculum sealing the throat is blown off as the pressure rises. Depending on the internal geometry, the ensuing pressure drop can sometimes cause rapid deformation of the propellant. In this paper we present the development of an experimental setup dedicated to the study of the 2D dynamic response of a solid propellant to propagating pressure waves. The experiment is first intended to assess whether the propellant’s response can be measured using various diagnostic techniques. The rectangular (parallelepiped‑shaped) chamber is slowly filled with air until the operculum bursts, causing the rapid depressurisation of the chamber. The resulting pressure wave travels through the chamber, impinging a specific shape of the propellant load, and induces oscillation as the wave reflects off the walls. The results show that the propellant obstacle, made of a conventional AP/HTPB research composition, oscillates in response to the pressure excitation as well as its internal mechanical forces. The test campaign is carried out at various pressure levels by adjusting the operculum thickness and involves complementary measurement techniques, such as pressure sensors that allow us to track the wave travelling through the chamber, while strain gauges measure the deformation directly on the propellant obstacle. Specific optical diagnostics (digital image correlation and point-tracking) are performed through side‑mounted windows. The setup and the measurement methods give very promising results for the characterisation of the dynamic behaviour of solid propellants during ignition, offering useful data for the development of mechanical models of the solid propellant. (10.60711/SPC2026.20260603.225861734990091247)
  DOI : 10.60711/SPC2026.20260603.225861734990091247
• Modélisation biomécanique des muscles extra-oculaires appliqué à un modèle optique
  
  • Bonnafé Julien
  • Allain Jean-Marc
  • Rio David
  , 2026. Nous développons un modèle éléments finis de l’œil, incluant le globe oculaire, les muscles extra-oculaires, le nerf optique et la graisse orbitale, afin d’analyser les efforts mécaniques lors de mouvements oculaires. Nous utilisons un modèle de Hill pour les muscles. Notre modèle inclut des contraintes internes (pression intraoculaire...). Au travers d’une géométrie simplifiée, nous montrons l’importance des contraintes internes et de la prise en compte de la graisse pour reproduire des mouvements réalistes de l’œil, et leurs conséquences sur la réfraction.
• Weak solutions of stochastic volterra equations in convex domains with general kernels
  
  • Abi Jaber Eduardo
  • Alfonsi Aurélien
  • Szulda Guillaume
  , 2026. We establish new weak existence results for d-dimensional Stochastic Volterra Equations (SVEs) with continuous coefficients and possibly singular one-dimensional nonconvolution kernels. These results are obtained by introducing an approximation scheme and showing its convergence. A particular emphasis is made on the stochastic invariance of the solution in a closed convex set. To do so, we extend the notion of kernels that preserve nonnegativity introduced in Alfonsi (2025) to non-convolution kernels and show that, under suitable stochastic invariance property of a closed convex set by the corresponding Stochastic Differential Equation, there exists a weak solution of the SVE that stays in this convex set. We present a family of non-convolution kernels that satisfy our assumptions, including a nonconvolution extension of the well-known fractional kernel. We apply our results to SVEs with square-root diffusion coefficients and non-convolution kernels, for which we prove the weak existence and uniqueness of a solution that stays within the nonnegative orthant. We derive a representation of the Laplace transform in terms of a non-convolution Riccati equation, for which we establish an existence result. (10.48550/arXiv.2506.04911)
  DOI : 10.48550/arXiv.2506.04911
• Learning Directional Randers Geometry for Latent-Space Exploration
  
  • Blanchard Théau
  • Allassonnière Stéphanie
  , 2026. Deep neural networks extract powerful latent representations from images, enabling fast and efficient downstream analysis. However, these latent spaces lack explicit geometric structure, leading to possibly poor interpolation fidelity and misleading extrapolation. Latent interpolation techniques assume symmetric transition costs which are often not the case for real world data. Their symmetric nature cannot capture directional processes, such as disease progression, where certain transitions are impossible (e.g. tumor growth dynamics or cognitive improvement in Alzheimer's disease). Although Riemannian metrics extend Euclidean geometry by incorporating curvature, they remain symmetric. We show that understanding asymmetry and not curvature alone directly addresses these problems. To do so we propose a new Finsler-Randers metric learning framework. It incorporates direction-dependent distances, thus enabling the computation of trajectories respecting monotonic sequential constraints. We demonstrate its potential for interpolation and extrapolation on controlled cross-sectional datasets.
• Geometry of Relaxed Fair Regression: A Unified Framework for Aware and Unaware Settings
  
  • Generali Lince Marie
  • Divol Vincent
  • Flamary Rémi
  • Gaucher Solenne
  • Loiseau Patrick
  , 2026. Fairness-accuracy trade-offs are a central concern in the deployment of fairness-aware machine learning methods. When sensitive attributes are unavailable at inference time–the so called unawareness setting, principled methods for obtaining accurate predictions under relaxed fairness constraints are largely missing. In this work, we address this gap by formulating regression under a demographic parity penalty as an optimal transport problem. Our framework unifies both the \emph{aware} and \emph{unaware} settings and characterizes optimal prediction functions via optimal transport maps, under both squared Wasserstein-2 and Total Variation penalties. These results reveal that the choice of penalty reflects fundamentally different fairness philosophies: the Wasserstein penalty induces a smooth, population-wide compromise, while Total Variation enforces exact parity for a subset of individuals. Building on these theoretical characterizations, we propose an algorithm that is simple to implement, computationally efficient, and consistently matches or outperforms state-of-the-art baselines on real-world benchmarks.
• POT Python Optimal Transport
  
  • Flamary Rémi
  • Vincent-Cuaz Cédric
  • Courty Nicolas
  • Gramfort Alexandre
  • Kachaiev Oleksii
  • Quang Tran Huy
  • David Laurène
  • Bonet Clément
  • Cassereau Nathan
  • Gnassounou Theo
  • Tanguy Eloi
  • Delon Julie
  • Collas Antoine
  • Mazelet Sonia
  • Chapel Laetitia
  • Kerdoncuff Tanguy
  • Yu Xizheng
  • Feickert Matthew
  • Krzakala Paul
  • Liu Tianlin
  • Fernandes Montesuma Eduardo
  , 2026. (10.5281/ZENODO.17161062)
  DOI : 10.5281/ZENODO.17161062
• A class of optimal virtual fields for inverse problems in elasticity
  
  • Chibli Nagham
  • Genet Martin
  • Imperiale Sébastien
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2026, 354 (G1), pp.417-449. This work addresses the identification of nonhomogeneous constitutive parameters from full-field measurements in both linear and nonlinear elasticity, considering incompressible as well as compressible materials. The inverse identification procedure relies on the Virtual Fields Method (VFM), which is based on the principle of virtual work with specifically chosen virtual fields. We propose an optimal class of virtual fields, designed to optimize the reconstruction stability with respect to measurement noise. A series of numerical experiments illustrate the effectiveness of the proposed approach. The method exhibits moderate sensitivity to measurement noise and remains robust even when the boundary conditions are only partially known. (10.5802/crmeca.361)
  DOI : 10.5802/crmeca.361
• Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing
  
  • Bonet Clément
  • Cazelles Elsa
  • Drumetz Lucas
  • Courty Nicolas
  , 2026. The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability distributions, it is natural to study this function in the Wasserstein space, which carries a rich formal Riemannian structure induced by Optimal Transport metrics. In this work, we investigate the existence and computation of Busemann functions in Wasserstein space, which admits geodesic rays. We establish closed-form expressions in two important cases: one-dimensional distributions and Gaussian measures. These results enable explicit projection schemes for probability distributions on $\mathbb{R}$, which in turn allow us to define novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets. We demonstrate the efficiency of those original schemes on synthetic datasets as well as transfer learning problems. (10.48550/arXiv.2510.04579)
  DOI : 10.48550/arXiv.2510.04579
• On Global Convergence Rates for Federated Softmax Policy Gradient Under Heterogeneous Environments
  
  • Labbi Safwan
  • Mangold Paul
  • Tiapkin Daniil
  • Moulines Eric
  , 2026, 300, pp.4960-4968. Ensuring convergence of policy gradient methods in federated reinforcement learning (FRL) under environment heterogeneity remains a major challenge. In this work, we first establish that heterogeneity, perhaps counter-intuitively, can necessitate optimal policies to be non-deterministic or even time-varying, even in tabular environments. Subsequently, we prove global convergence results for federated policy gradient (FedPG) algorithms employing local updates, under a Łojasiewicz condition that holds only for each individual agent, in both entropy-regularized and non-regularized scenarios. Crucially, our theoretical analysis shows that FedPG attains linear speed-up with respect to the number of agents, a property central to efficient federated learning. Leveraging insights from our theoretical findings, we introduce b-RS-FedPG, a novel policy gradient method that employs a carefully constructed softmax-inspired parameterization coupled with an appropriate regularization scheme. We further demonstrate explicit convergence rates for b-RS-FedPG toward near-optimal stationary policies. Finally, we demonstrate that empirically both FedPG and b-RS-FedPG consistently outperform federated Q-learning on heterogeneous settings.
• Regularity of solutions of the Dyson equation and applications
  
  • Pesce Valentin
  , 2026. The goal of this short paper is to investigate the regularity of the solutions of the Dyson equation. In the work of Bertucci and al. [3, 4, 5], a new notion of solutions for the Dyson equation has been introduced using the viscosity solutions theory and they proved a regularization of solutions in L^{∞} . According to the work of Biane [6] we should expect a regularization in C^{1/3} of solutions (and not better). In the spirit of the works of Bertucci and al., we shall prove using PDE methods that for almost all time t \ge 0 the solution is as expected in C^{1/3} . Our approach allows us to extend this result in addition to a drift term in the Dyson equation for which the semi explicit solutions is not necessarily known. We shall also give an application of this result, proving that the solutions of the periodic Dyson equation converge in long-time toward the uniform distribution on the circle in L^{∞} norm which was an open question in [5].
• A characterization of Generalized functions of Bounded Deformation
  
  • Chambolle Antonin
  • Crismale Vito
  Journal of Functional Analysis, Elsevier, 2026, 290 (9), pp.111391. <div><p>We show that Dal Maso's GBD space, introduced for tackling crack growth in linearized elasticity, can be defined by simple conditions in a finite number of directions of slicing.</p></div> (10.1016/j.jfa.2026.111391)
  DOI : 10.1016/j.jfa.2026.111391
• Numerical approximation of ergodic BSDEs using non linear Feynman-Kac formulas
  
  • Gobet Emmanuel
  • Richou Adrien
  • Szpruch Lukasz
  Stochastic Processes and their Applications, Elsevier, 2026, 195, pp.104871. In this work we study the numerical approximation of a class of ergodic Backward Stochastic Differential Equations. These equations are formulated in an infinite horizon framework and provide a probabilistic representation for elliptic Partial Differential Equations of ergodic type. In order to build our numerical scheme, we put forward a new representation of the PDE solution by using a classical probabilistic representation of the gradient. Then, based on this representation, we propose a fully implementable numerical scheme using a Picard iteration procedure, a grid space discretization and a Monte-Carlo approximation. Up to a limiting technical condition that guarantees the contraction of the Picard procedure, we obtain an upper bound for the numerical error. We also provide some numerical experiments that show the efficiency of this approach for small dimensions. (10.1016/j.spa.2026.104871)
  DOI : 10.1016/j.spa.2026.104871
• Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees
  
  • Silveri Marta Gentiloni
  • Conforti Giovanni
  • Durmus Alain
  International Conference on Machine Learning, Proceedings of Machine Learning Research, 2026. Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood. In this work, we provide refined and novel convergence guarantees for Brownian motion based DFMs, focusing on the discretization error. Our analysis is conducted under the Kullback-Leibler (KL) divergence and the 2-Wasserstein distance. Under finite-moment conditions and a mild score integrability assumption, we derive KL convergence bounds with improved dimensional dependence compared to prior work, achieving, up to our knowledge, state-of-the-art scaling under minimal conditions. We further extend the analysis to the 2-Wasserstein distance: under an additional first-order score integrability assumption and a weak log-concavity condition, we obtain convergence guarantees with dimensional dependence consistent with the KL case.
• A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings
  
  • Alasseur Clémence
  • Bayraktar Erhan
  • Dumitrescu Roxana
  • Jacquet Quentin
  , 2026. In this paper, we consider the problem of a Principal aiming at designing a reward function for a population of heterogeneous agents. We construct an incentive based on the ranking of the agents, so that a competition among the latter is initiated. We place ourselves in the limit setting of mean-field type interactions and prove the existence and uniqueness of the equilibrium distribution for a given reward, for which we can find an explicit representation. Focusing first on the homogeneous setting, we characterize the optimal reward function using a convex reformulation of the problem and provide an interpretation of its behaviour. We then show that this characterization still holds for a sub-class of heterogeneous populations. For the general case, we propose a convergent numerical method which fully exploits the characterization of the mean-field equilibrium. We develop a case study related to the French market of Energy Saving Certificates based on the use of realistic data, which shows that the ranking system allows to achieve the sobriety target imposed by the regulation.
• Malliavin calculus for signatures with applications to finance
  
  • Abi Jaber Eduardo
  • Rey Clément
  • Sotnikov Dimitri
  , 2026. Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As a consequence, we obtain closed-form expressions for the Clark-Ocone representation, the Ornstein-Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.
• PSDNORM: TEST-TIME TEMPORAL NORMALIZATION FOR DEEP LEARNING IN SLEEP STAGING
  
  • Gnassounou Theo
  • Collas Antoine
  • Flamary Rémi
  • Gramfort Alexandre
  , 2025. <div><p>Distribution shift poses a significant challenge in machine learning, particularly in biomedical applications using data collected across different subjects, institutions, and recording devices, such as sleep data. While existing normalization layers, BatchNorm, LayerNorm and InstanceNorm, help mitigate distribution shifts, when applied over the time dimension they ignore the dependencies and auto-correlation inherent to the vector coefficients they normalize. In this paper, we propose PSDNorm that leverages Monge mapping and temporal context to normalize feature maps in deep learning models for signals. Notably, the proposed method operates as a test-time domain adaptation technique, addressing distribution shifts without additional training. Evaluations with architectures based on U-Net or transformer backbones trained on 10K subjects across 10 datasets, show that PSDNorm achieves state-of-the-art performance on unseen left-out datasets while being 4-times more data-efficient than BatchNorm.</p></div> (10.48550/arXiv.2503.04582)
  DOI : 10.48550/arXiv.2503.04582
• Neural Optimal Transport Meets Multivariate Conformal Prediction
  
  • Kondratyev Vladimir
  • Fishkov Alexander
  • Hegazy Mahmoud
  • Kotelevskii Nikita
  • Flamary Rémi
  • Panov Maxim
  • Moulines Eric
  , 2026, pp.77879–77917.
• Numerical approximation of Markovian BSDEs in infinite horizon and elliptic PDEs
  
  • Gobet Emmanuel
  • Richou Adrien
  • Shardul Charu
  , 2026. <div><p>We study backward stochastic differential equations (BSDEs) in infinite horizon and design efficient numerical schemes for solving them. We establish a probabilistic representation of the solution of the BSDE using Malliavin derivative and prove results for contraction of a Picard scheme. We develop three numerical schemes, of which the first two are based on a fixed point argument using contraction, imposing additional assumptions compared to what is needed for existence and uniqueness of the solution. The first scheme is a space grid based approximation where we establish tight numerical error bounds using a growth truncation argument; it performs well in low dimensions but computational times increase exponentially with dimension. The second scheme uses neural network approximations for which we have proved a convergence result. Using neural networks alleviates the curse of dimensionality, giving good accuracy in very high dimensions. The third scheme also uses neural networks but does not rely on contraction arguments, showcasing good performance even for larger z-Lipschitz dependence outside the domain of contraction.</p></div>
• Online Decision-Focused Learning
  
  • Capitaine Aymeric
  • Haddouche Maxime
  • Moulines Eric
  • Jordan Michael I.
  • Boursier Etienne
  • Durmus Alain
  , 2026, pp.46821–46854. Decision-focused learning (DFL) is an increasingly popular paradigm for training predictive models whose outputs are used in decision-making tasks. Instead of merely optimizing for predictive accuracy, DFL trains models to directly minimize the loss associated with downstream decisions. However, existing studies focus solely on scenarios where a fixed batch of data is available and the objective function does not change over time. We instead investigate DFL in dynamic environments where the objective function and data distribution evolve over time. This setting is challenging for online learning because the objective function has zero or undefined gradients, which prevents the use of standard first-order optimization methods, and is generally non-convex. To address these difficulties, we (i) regularize the objective to make it differentiable and (ii) use perturbation techniques along with a near-optimal oracle to overcome non-convexity. Combining those techniques yields two original online algorithms tailored for DFL, for which we establish respectively static and dynamic regret bounds. These are the first provable guarantees for the online decision-focused problem. Finally, we showcase the effectiveness of our algorithms on a knapsack experiment, where they outperform two standard benchmarks.
• Asymptotic Stability of Equilibrium States for Cohesive Fluids
  
  • Giovangigli Vincent
  , 2026. We investigate asymptotic stability of constant equilibrium states for compressible nonisothermal cohesive fluids also termed capillary fluids or diffuse interface fluids. The density gradient is added as an extra variable and the augmented system of equations is recast into a normal form with symmetric transport first order terms, symmetric dissipative second order terms and antisymmetric cohesive second order terms. Global existence and asymptotic stability of constant equilibrium states are established by using new dissipative conditions for such augmented hyperbolic-parabolic-dispersive systems of equations. Decay estimates are obtained in all spatial dimensions by using the augmented formulation as well as estimates in Fourier spaces.
• Efficient Monte Carlo sampling of metastable systems using nonlocal collective variable updates
  
  • Schönle Christoph
  • Carbone Davide
  • Gabrié Marylou
  • Lelièvre Tony
  • Stoltz Gabriel
  The Journal of Chemical Physics, American Institute of Physics, 2026, 164 (15), pp.154107. Monte Carlo simulations are widely used to simulate complex molecular systems, but standard approaches suffer from metastability. Lately, the use of nonlocal proposal updates in a collective-variable (CV) space has been proposed in several works. Here, we generalize these approaches and explicitly spell out an algorithm for nonlinear CVs and underdamped Langevin dynamics. We prove reversibility of the resulting scheme and demonstrate its performance on several numerical examples, observing a substantial performance increase compared to methods based on overdamped Langevin dynamics as considered previously. Advances in generative machine-learning-based proposal samplers now enable efficient sampling in CV spaces of intermediate dimensionality—tens to hundreds of variables—and our results extend their applicability toward more realistic molecular systems. (10.1063/5.0318016)
  DOI : 10.1063/5.0318016
• Growing random planar network with oriented branching and fusion
  
  • Bansaye Vincent
  • Raoul Gaël
  • Tomasevic Milica
  , 2026. We consider a growing planar network where a tip grows at constant speed, branches at constant rate and inactivates when it meets a branch already created. We only consider here orthogonal branching occurring always in the same direction. This yields a spatial branching property to the growing network. The connected components of the network then form a branching process of rectangles with double immigration. Using a spine approach for a typical rectangle and coupling arguments, the study is boiled down to a one dimensional stick breaking model with aging. We can then prove long time convergence of empirical measure of the family of rectangles after polynomial rescaling. The limiting distribution and speed of convergence can be explicitly described. The proofs also rely on the description of common ancestor of rectangles in the branching structure with double immigration.
• PSWF-Radon approach to reconstruction from band-limited Hankel transform
  
  • Goncharov Fedor
  • Isaev Mikhail
  • Novikov Roman
  • Zaytsev Rodion
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2026. We give new formulas for reconstructions from band-limited Hankel transform of integer or half-integer order. Our formulas rely on the PSWF-Radon approach to super-resolution in multidimensional Fourier analysis. This approach consists of combining the theory of classical one-dimensional prolate spheroidal wave functions with the Radon transform theory. We also use the relation between Fourier and Hankel transforms and Cormack-type inversion of the Radon transform. Finally, we investigate numerically the capabilities of our approach to super-resolution for band-limited Hankel inversion in relation to varying levels of noise. (10.1515/jiip-2025-0058)
  DOI : 10.1515/jiip-2025-0058
• From stochastic individual-based models to free-boundary Hamilton-Jacobi equations
  
  • Champagnat Nicolas
  • Méléard Sylvie
  • Mirrahimi Sepideh
  • Tran Chi
  , 2026. We study a stochastic branching model for a population structured by a quantitative phenotypic trait and subject to births, deaths, and mutations. In a regime of large population and small mutations, and in logarithmic scales of size and time, we derive a certain class of free boundary Hamilton-Jacobi equations with state constraints from the stochastic individual-based system. This goes beyond the classical Hamilton-Jacobi equations obtained from deterministic models by taking into account the possible extinction of the system in certain regions of the trait space. The proof is obtained by combining methods for the analysis of Hamilton-Jacobi equations with probabilistic tools from the theory of large deviations and branching processes.