Publications

2026

• 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.
• 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
• 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.
• 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
• 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
• 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].
• 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.
• 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.
• 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.
• 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.
• 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.
• 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
• 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.
• Acute Smurf mortality and age-dependence in a two-phase ageing model.
  
  • Breuil Luce
  • Doumic Marie
  • Kaakaï Sarah
  • Rera Michaël
  , 2026. Ageing is traditionally conceived as a continuous process of progressive physiological decline. However, recent evidence across species suggests that ageing may instead proceed through distinct phases. Using state-of-the-art statistical methods, we develop a rigorous analysis of longitudinal survival data from 1,159 individually tracked female Drosophila melanogaster. This data-driven analysis leads us to introduce a new parametric model of transition rates within the two-phase ageing framework. Flies were monitored using the Smurf assay, which detects increased intestinal permeability through leakage of an ingested blue dye, and is a strong biological marker of ageing. The Smurf phenotype identifies a sharp transition from a non-Smurf state to a Smurf state that precedes death. Our results yield three key findings. First, the Smurf transition rate follows a Gompertz-Makeham law, increasing exponentially with age. Second, contrary to previous constant-rate assumptions, newly transitioned Smurf flies exhibit remarkably high mortality - approximately 40% die within 24 hours - followed by an exponential decline in death rate that stabilises to a lower constant baseline. Third, we identified a mild but statistically significant negative dependence between time spent non-Smurf and subsequent Smurf lifespan. Our best-fit model captures a potential bimodal nature of mortality curves using simple, biologically interpretable functions. Validation using data from two mouse strains confirms the broader applicability of this framework. These results establish a quantitative foundation for the two-phase ageing paradigm and highlight a critical period of vulnerability immediately following the physiological transition to frailty. (10.64898/2026.02.18.706552)
  DOI : 10.64898/2026.02.18.706552
• Scattering and inverse scattering for multipoint potentials at high energies
  
  • Kuo Pei-Cheng
  • Novikov Roman
  , 2026. We consider the Schrödinger equation with a multipoint potential of Bethe-Peierls-Thomas-Fermi type. For this singular potential, we develop scattering and inverse scattering at high energies. In particular, in this framework, our results include analogs of the "regular" Born-Faddeev formula for the scattering amplitude and analogs of related "regular" inverse scattering reconstructions at high energies. Related results for scattering solutions at high energies are also presented.
• Pareto Set Characterization in Constrained Multiobjective Optimization and the COBI Problem Generator
  
  • Auger Anne
  • Brockhoff Dimo
  • Opravš Luka
  • Tušar Tea
  , 2026. Benchmark problems play a central role in assessing the performance of numerical optimization algorithms. However, many existing constrained multiobjective optimization benchmark problems rely on overly restricted constructions or lack formal analysis of their optimal solution sets, limiting their relevance for systematic algorithm evaluation. In this work, we introduce a class of analytically tractable constrained multiobjective optimization problems whose Pareto sets can be formally characterized. The construction is based on convex-quadratic functions with positive definite Hessians, combined through multipeak formulations in which each objective is defined as the minimum over several convex-quadratic components. This approach preserves analytical structure while enabling multimodality (non-convexity), ill-conditioning and non-separability. The constraints are built as sublevel sets of multipeak functions giving rise to problems with potentially disconnected feasible regions. Building on these results, we propose COBI, a scalable generator of constrained bi-objective test problems designed for benchmarking derivative-free optimization algorithms. We provide a reference Python implementation that enables straightforward integration of COBI instances into benchmarking workflows.
• Error-Based mesh selection for efficient numerical simulations with variable parameters
  
  • Dornier Hugo
  • Le Maître Olivier P
  • Congedo Pietro Marco
  • Salah El Din Itham
  • Marty Julien
  • Bourasseau Sébastien
  Computers and Fluids, Elsevier, 2026, 313. Advanced numerical simulations often depend on mesh refinement techniques to manage discretization errors in complex models and reduce computational costs. This work concentrates on Adaptive Mesh Refinement (AMR) for steady-state solutions, which uses error estimators to iteratively refine the mesh locally and gradually tailor it to the solution. AMR requires evaluating the solution across a series of meshes. When solving the model for multiple operating conditions, such as in uncertainty quantification studies, full systematic adaptation can cause significant computational overhead. To mitigate this, the Error-based Mesh Selection (EMS) method is introduced to decrease the cost of adaptation. For each operating condition, EMS seeks to choose, from a library of pre-adapted meshes, the one that minimizes the discretization error. A key feature of this approach is the use of Gaussian Process models to predict the solution errors for each mesh in the library. These error models are built solely from the library's meshes and their solutions, using restriction errors as proxies for discretization errors, thereby avoiding additional model evaluations. The EMS method is tested on an analytical shock problem and a supersonic scramjet configuration, showing near-optimal mesh selection. The influence of library size on the resulting error level is also examined. (10.1016/j.compfluid.2026.107081)
  DOI : 10.1016/j.compfluid.2026.107081
• Molecular Scattering Distributions under New Boundary Conditions for the Boltzmann Equation
  
  • Kosuge Shingo
  • Aoki Kazuo
  • Giovangigli Vincent
  • Golse François
  , 2026. The scattering properties of kinetic boundary conditions for the Boltzmann equation recently proposed by Aoki et al. [see, e.g., Phys. Rev. E 106, 035306 (2022)] are investigated. Scattering patterns of reflected molecules are obtained for molecular-beam and Maxwellian incident distributions, and representative numerical results are presented.
• Robust topology optimization accounting for uncertain micro-structural changes in metallic additive manufacturing
  
  • Masson H.
  • Denimal Goy Enora
  • Peigney Michaël
  Structural and Multidisciplinary Optimization, Springer Verlag, 2026, 69 (4), pp.103. Abstract Additive Manufacturing (AM) can introduce defects in the material or variations in its properties, which can alter the mechanical performance of 3D-printed parts and lead to discrepancies between predicted and observed behavior. To mitigate this effect, it is essential to incorporate these variabilities into the design process to ensure optimal structural performance. Topology Optimization (TO) methods determine the optimal distribution of material within a design domain and often result in complex parts. AM is particularly relevant for manufacturing such parts. To reach robust designs, it is essential to consider material variability arising from AM in TO. Rather than relying on computationally expensive simulations to characterize these material variations, this study presents a new, practical, simplified, parametrized stochastic model that accounts for these variations in the AM process. This stochastic modeling directly optimizes performance in terms of average and variability. The work specifically focuses on variations in material properties along the build direction. More specifically, it addresses the transition from columnar grains (which have anisotropic properties) to equiaxed grains (which have isotropic properties) in metallic AM. This transforms the topology optimization problem into a robust topology optimization problem, yielding an optimal shape that reduces variability related to the AM process. TO is performed using the SIMP method. Gaussian quadrature is employed to reduce the computational cost of the statistical moments of the objective function. The influence of individual and combined transition stochastic model parameters is analyzed. The results demonstrate that the robust geometry systematically outperforms the deterministic ones, showing the effectiveness of the method in reducing performance variability. More specifically, for similar average properties, variability is substantially reduced. These findings support the integration of robust design strategies with AM. (10.1007/s00158-026-04279-w)
  DOI : 10.1007/s00158-026-04279-w
• Do you precondition on the left or on the right?
  
  • Spillane Nicole
  • Matalon Pierre
  • Szyld Daniel B
  , 2026. This work is a follow-up to a poster that was presented at the DD29 conference. Participants were asked the question: “Do you precondition on the left or on the right?”. Here we report on the results of this social experiment. We also provide context on left, right and split preconditioning, share our literature review on the topic, and analyze some of the finer points. Two examples illustrate that convergence bounds can sometimes lead to misleading conclusions.