Publications

2026

• Wiring the Fly Brain into a Hierarchical World Model A self-supervised JEPA world model of a complete connectome Team Piaget
  
  • Thil Lucas
  • Nowak Assis Daniel
  • Berthier Louis
  • Jlidi Adam
  , 2026. World models predict the future of an environment in an abstract latent space, the kind of prediction a brain is thought to perform. We ask whether such a model can learn the dynamics of a real brain rather than a metaphor for one. We take the FlyWire connectome of the adult Drosophila, the only animal whose brain is fully mapped (∼139,000 neurons, ∼50M synapses), turn it into a spiking brain model embodied in a physics simulation of the fly's body, and let it act in a closed sensorimotor loop. On the resulting whole-brain spike trains we train a hierarchy of region-level world models (Joint-Embedding Predictive Architectures, JEPAs), one per brain region, coupled along the connectome's own sensory → integration → motor flow. To our knowledge this is the first JEPA world model of a complete brain. We report three findings: the motor (descending) pathway behaves as a genuine world model, forecasting its own activity and linearly decoding real motor commands; the extreme sparsity of spike codes induces a characteristic latent collapse, diagnosable by participation ratio, which a connectome-derived encoder helps counteract; and coupling regions along the connectome's feedforward structure measurably improves prediction, with the gain growing over longer horizons. The brain's wiring does not merely inspire the architecture, it improves the model of the world.
• Nonequilibrium and multiscale fluids: Models and approximations
  
  • Pichard Teddy
  , 2026.
• Optimized high-order IMEX-RK schemes for degenerate diffusion-reaction problems with application to travelling waves phenomena
  
  • Antonietti Paola F.
  • Corti Mattia
  • Orlando Giuseppe
  , 2026. We study a class of IMplicit-EXplicit Runge--Kutta (IMEX-RK) schemes for the numerical approximation of reaction and diffusion-reaction problems arising in a variety of biological and physical applications. Such models may admit travelling wave solutions, with the Fisher--Kolmogorov equation representing a prototypical example. Motivated by this feature, the proposed time integration schemes are designed to accurately capture sharp propagating fronts. We also investigate a less standard use of IMEX-RK methods that circumvents a splitting of reaction terms into linear and nonlinear components, while still requiring the solution of linear systems at each stage. This semi-implicit formulation, referred to as SI-IMEX-RK, enables a targeted treatment of stiffness by isolating its relevant contributions. The time discretization is coupled with a high-order polygonal discontinuous Galerkin method for space discretization, resulting in a flexible and robust framework for the treatment of multiscale dynamics in complex geometries. A comprehensive validation strategy is presented to assess the accuracy and stability properties of the proposed schemes across a hierarchy of increasingly challenging test problems.
• Body-fitted topology optimization for multi-material electrical machine design
  
  • Gauthey Thomas
  , 2026. Achieving carbon neutrality by 2050 is a major challenge of the 21st century. Meeting this target requires rethinking both technology and usage patterns. The transport sector currently accounts for 28.5% of these emissions. Aviation contributes 4%, mainly from intercontinental long-haul flights. Decarbonising the sector is necessary for its long-term viability. This can be achieved through large-scale electrification and increased on-board power density. Electric machines tailored to the sector's needs, advanced by additive manufacturing, are critical. Applications include hybridisation, energy recovery, or replacing pneumatic actuators. Their design represents a major technological challenge.This thesis focuses on the development of tools and methods for topological optimisation based on the level-set method, with conformal remeshing, thereby enabling an efficient exploration of the design space for electric actuators.The thesis is divided into four main chapters. It begins with a review of the literature on optimal design methods. These range from conventional parametric optimisation, which uses a limited number of variables and offers little potential for innovation, to topological optimisation. Topological optimisation treats the entire shape as the optimisation variable, helping overcome design biases. An overview of the types of machines explored in this thesis is then provided. These include synchro-reluctance machines, with or without permanent magnet assistance, covering both their modelling and design.The second chapter proposes a methodological framework for body-fitted levet-set topological optimisation. Following an introduction to shape sensitivity calculations using Hadamard's boundary variation method—the core of the structure evolution algorithm—and a standard example for magnetic topological optimisation, the specific features of optimisation using the body-fitted level-set method are presented. An exploration of a certain class of optimisation criteria (including dual-energy) known as self-adjoint criteria for a selected physical problem is undertaken; these criteria offer significant potential to reduce computation time, as their sensitivity computation does not require an adjoint PDE.The third chapter formalises and implements criteria for optimizing electric machines. These criteria may be purely geometric, such as volume or moment of inertia, or may depend on mechanical partial differential equations, such as the von Mises criterion. Others depend on magnetic PDEs, such as magnetic flux or magnetic torque. Attention is paid to the design space boundaries that the algorithm must identify to achieve visible salience. As in Chapter Two, a more theoretical perspective on global magnetic forces is proposed. This is based on the self-adjoint nature of dual energy. A formal equivalence is suggested between various formulas in the literature and the force formula obtained by formal differentiation.The fourth and final chapter of the manuscript deals with single- and multi-material design for electrical machines, focusing on their specific characteristics, including rotational motion and the evaluation of criteria that require the collection of magnetic states, such as average torque. Multi-material design is then addressed by adding a second level-set function to enable the optimisation of a non-magnetic steel phase. Finally, a coupling with parametric optimisation to integrate magnets into the design is then explored, all with conformal multi-phase remeshing as a backdrop.
• Mean field games with incomplete information
  
  • Bertucci Charles
  , 2026. This paper is concerned with mean field games in which the players do not know the repartition of the other players. First a case in which the players do not gain information is studied. Results of existence and uniqueness are proved and discussed. Then, a case in which the players observe the payments is investigated. A master equation is derived and partial results of uniqueness are given for this more involved case.
• Multi-fluid modelling of two-phase flows : non-equilibrium models, two-scale approaches and numerical simulations
  
  • Haegeman Ward
  , 2026. Two-phase flows occur in a wide range of industrial applications, particularly in the defence and aerospace industries. Numerical simulations have become essential design tools; however, the computational cost can become prohibitive for flows involving numerous physical phenomena across a wide range of scales. The development of reduced-order models therefore appears to be a promising strategy. Among the various possible approaches, multi-fluid approaches allow the development of models suited to different flow regimes, and at different modelling levels.The aim of this thesis is to develop multi-fluid models to describe high-velocity compressible two-phase flows. As the objective is to enable the design of predictive numerical tools, particular attention is paid to the mathematical structure of the PDEs and the associated numerical schemes.In the first part, we focus on a specific class of multi-fluid models and identify the appropriate level of modelling required for the targeted applications. A two-temperature multi-fluid model is identified and its numerical strategy is based on instantaneous pressure relaxation techniques. The study of the scheme and the development of new relaxation techniques then enable robust numerical simulations of high-velocity compressible two-phase flows.However, these models do not allow us to describe all the relevant scales of the flow. Consequently, in the second part, we develop a two-scale model. The proposed model is derived using variational principles and irreversible thermodynamics. Its source terms allow for a transfer between scales and for a regularisation of the large-scale interface.Finally, in the third part, we develop a multi-fluid model that allows the phases to be fully out of equilibrium. Derived through variational principles, it meets both the requirements of physical modelling and the mathematical criteria necessary to correctly handle shock-type solutions.
• GeoEpidemio: combining 2D and 3D spatio-temporal visualizations to support simulated epidemiological data analysis
  
  • Lobo M-J
  • Gautier J.
  • Colomb Maxime
  • Dutremble A.
  • Choqueux G.
  , 2026. The COVID-19 outbreak rapidly spread worldwide in 2020, prompting diverse public health policies with varying outcomes. Understanding and analyzing the spatio-temporal dynamics of virus propagation is crucial for assessing interventions and preparing for future outbreaks. Epidemiological simulations can model virus spread under different policy scenarios, but their outputs are highly detailed and challenging to visualize. We present GeoEpidemio, a spatio-temporal visualization system that combines 2D and 3D representations to explore simulated epidemic data. We demonstrate its effectiveness through a case study comparing multiple scenarios with different intervention strategies. (10.2312/evs.20261016)
  DOI : 10.2312/evs.20261016
• Sampling and Transport with Score-based Diffusion Models : Opportunities and Methodological Advances
  
  • Noble-Bourillot Maxence
  , 2026. This thesis investigates the design of new methodologies for sampling and optimal transport, primarily leveraging recently developed generative models known as score-based diffusion models. The first part addresses two fundamental challenges in sampling. We first consider geometrically constrained state spaces, for which we propose a novel Riemannian Hamiltonian Monte Carlo-based method that is proven to be unbiased. We then turn to the challenge of multi-modality, highlighting structural limitations shared by broad classes of existing approaches, including classical Monte Carlo methods. To tackle multi-modality difficulty, the second part introduces diffusion models into sampling schemes and develops three complementary methodologies: a learning-free diffusion-based sampler relying solely on Monte Carlo estimators; an annealed variational inference diffusion-based sampler that uses informative guidance to mitigate mode collapse; and a new class of Boltzmann Generators combining diffusion models with annealed Monte Carlo techniques. In the final part, we leverage diffusion models within the Schrödinger Bridge framework to address advanced optimal transport problems. We first introduce a method for solving multi-marginal optimal transport problems with tree-structured cost functions, with applications to Wasserstein barycenters. Then we develop an unbiased bridge-matching approach for numerically solving generalized Schrödinger Bridge problems motivated by the inference of population dynamics from snapshot longitudinal data.
• An inverse tsunami problem in the time domain: a well-posedness analysis of the forward problem and an inversion strategy based on a mixed formulation of the Tikhonov regularization
  
  • Bourgeois Laurent
  • Moireau Philippe
  • Terrine Raphaël
  , 2026. This contribution concerns an inverse problem related to a tsunami in the ocean, the tsunami being caused by a submarine earthquake. Considering the very beginning of the phenomenon, a simple linear model incorporating both gravity and acoustic waves is proposed. The main objective is to develop a strategy to solve the inverse problem of retrieving the bottom displacement from the induced free surface perturbation. Such strategy is based on a mixed formulation of the Tikhonov regularization in the space/time domain, the regularization parameter being determined by using the Morozov principle by means of duality in optimization. Some numerical experiments in 2D, which rely on a tensorized finite element method, show that our strategy is effective. A secondary objective is to prove existence and uniqueness of both strong and variational solutions to the forward problem.
• Numerical analysis of an optimal control approach to solve a tsunami inverse problem
  
  • Bourgeois Laurent
  • Moireau Philippe
  • Terrine Raphaël
  , 2026. This paper concerns the reconstruction of an abrupt bottom displacement of the ocean from the measurement of the induced perturbation of the free surface, which is a severely ill-posed inverse problem. This problem is solved by using an optimal control approach, the physics being governed by a time evolution system based on a simple oceanography model. We firstly recast the problem in an abstract framework, secondly propose an implicit Euler scheme for the time discretization combined with a Finite Element method for the space discretization. The main result is an error estimate between the solution to the discrete control optimal problem and the solution to the continuous optimal problem, which is obtained by considering the discrete and continuous weak mixed formulations that characterize the optimality for these two problems. Some numerical experiments illustrate the efficiency of our approach and the consistency of our error estimate.
• On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers
  
  • Durmus Alain
  • Gruffaz Samuel
  • Kailas Miika
  • Saksman Eero
  • Vihola Matti
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2026, 36 (3). There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution. (10.1214/25-AAP2269)
  DOI : 10.1214/25-AAP2269
• The Mortensen observer on the space of probability measures
  
  • Morange Martin
  , 2026. We study a deterministic filtering problem formulated directly on the Wasserstein space of probability measures with finite second moment. Motivated by the Mortensen minimum-energy observer, we consider the reconstruction of an evolving probability density from partial observations by minimizing an action functional combining a kinetic transport cost and a time-dependent observation mismatch. The resulting value function is defined on the infinite-dimensional manifold (P_2(R^d), W_2) and satisfies a Hamilton-Jacobi-Bellman equation involving the Wasserstein gradient. Under suitable regularity and growth assumptions on the observation functional, we establish dynamic programming principles, continuity of the value function, existence of minimizing trajectories, and viscosity solution properties of the associated Hamilton-Jacobi equation. We provide two complementary notions of viscosity solutions: a geometric formulation based on subdifferentials in Wasserstein space, and a Hilbertian formulation inspired by Lions' lifting approach. This allows us to prove a comparison principle and uniqueness of solutions. Extensions to transport equations with drift are also discussed. Finally, we introduce a semi-Lagrangian scheme in order to approximate the value function, and show Γ-convergence of the scheme.
• The Wasserstein cost of importance sampling
  
  • Coste Simon
  • Goldman Michael
  , 2026. Importance sampling (IS) consists in biasing samples from a distribution $f$ towards another distribution $g$. Concretely, given samples $X_i$ from $f$, the IS measure is $$\hat{g}_n = \frac{1}{Z_n}\sum_{i=1}^n \frac{g(X_i)}{f(X_i)} \delta_{X_i},$$ with $Z_n = \sum_{i=1}^n \frac{g(X_i)}{f(X_i)}$. The random measure $\hat{g}_n$ approximates $g$, and is used in many contexts ranging from Monte Carlo integration to Bayesian inference. We show that, in high dimension ($d \geqslant 3$), the Wasserstein cost $W_p^p(\hat{g}_n, g)$ has order $n^{-p/d}$ in expectation, i.e. $$\beta^{\mathrm{low}}_{p,d}\int gf^{-p/d}\leqslant \liminf_{n \to \infty} n^{p/d} \mathbb{E}[W_p^p(\hat{g}_n, g)] \leqslant \limsup_{n \to \infty} n^{p/d} \mathbb{E}[W_p^p(\hat{g}_n, g)] \leqslant\beta_{p,d} \int g f^{-p/d}$$ where $\beta^{\mathrm{low}}_{p,d}\leqslant \beta_{p,d}$ are constants depending only on $p$ and $d$, which are equal for $p=2$ and conjectured to be equal for any $p\geqslant 1$. Our results are valid for all $p\geqslant 1$ and $d\geqslant 3$. In the case where $\beta^{\mathrm{low}}_{p,d}= \beta_{p,d}$, we show that the asymptotically optimal sampling distribution $f^*$ for importance sampling is not equal to $g$ but to a tempered version of $g$, namely $f^* \propto g^{d/(p+d)}$, which is reminiscent of Zador’s theorem in the domain of measure quantization.
• Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES
  
  • Matalon Pierre
  • Spillane Nicole
  , 2026. The convergence of the GMRES linear solver is notoriously hard to predict. A particularly enlightening result by [Greenbaum, Pták, Strakoš, 1996] is that, given any convergence curve, one can build a linear system for which GMRES realizes that convergence curve. What is even more extraordinary is that the eigenvalues of the problem matrix can be chosen arbitrarily. We build upon this idea to derive novel results about weighted GMRES. We prove that for any linear system and any prescribed convergence curve, there exists a weight matrix M for which weighted GMRES (i.e. GMRES in the inner product induced by M ) realizes that convergence curve, and we characterize the form of M . Additionally, we exhibit a necessary and sufficient condition on M for the simultaneous prescription of two convergence curves, one realized by GMRES in the Euclidean inner product, and the other in the inner product induced by M . These results are then applied to infer some properties of preconditioned GMRES when the preconditioner is applied either on the left or on the right. For instance, we show that any two convergence curves are simultaneously possible for left and right preconditioned GMRES.
• Comparison Between Effective and Individual Growth Rates in a Heterogeneous Population
  
  • Doumic Marie
  • Rat Anaïs
  • Tournus Magali
  Royal Society Open Science, The Royal Society, 2026, 13 (5), pp.251119. Is there an advantage to heterogeneity in a population where individuals grow and divide by fission? This is a broad question, to which there is no easy universal answer. This article aims to provide a quantitative answer in the specific context of growth rate heterogeneity by comparing the fitness of homogeneous versus heterogeneous populations. We focus on size-structured populations, where the growth rate of each individual is set at birth by heredity and/or random mutations. The fitness (or Malthus parameter, or effective fitness) of such heterogeneous population is defined by its long-term behaviour, and we introduce the effective growth rate as the individual growth rate in the homogeneous population with the same fitness. We derive analytical formulae linking effective and individual growth rates in two paradigmatic cases: first, constant growth and division rates, second, linear growth rates and uniform fragmentation. Surprisingly, these two cases yield similar expressions. Then, by comparing the fitness and the effective growth rates of populations with different degrees of heterogeneity or different laws of heredity/mutation to those of average homogeneous populations, we quantitatively investigate the combined influence of heredity and heterogeneity, and revisit previous results stating that heterogeneity is beneficial in the case of strong heredity. (10.1098/rsos.251119)
  DOI : 10.1098/rsos.251119
• Volterra clocks and their pure-jump limits: hitting times of curved boundaries
  
  • Abi Jaber Eduardo
  • Attal Elie
  • Søjmark Andreas
  , 2026. We introduce a class of continuous Volterra processes, called Volterra clocks, and study their singular limit as the memory kernel collapses to a Dirac mass at zero. The dynamics are parametrised by a function f acting as a nonlinear time- change, generalising the Volterra square-root process and recovering it when f is affine. In the singular limit, the continuous Volterra clock converges weakly to a pure-jump process given by first passage times of a Brownian motion to curved boundaries, including affine and square-root boundaries when f is, respectively, affine or quadratic. Outside the affine setting, characteristic function methods are no longer available, and we instead identify the limit directly from the dynamics. We do this through a topological framework adapted to the time-change structure which involves Skorokhod’s M1 topology and a decorated notion of convergence. Our analysis unifies several regimes of interest for general Volterra clocks, including large-time asymptotics, fast mean reversion, and hyper-roughness. In particular, this subsumes and extends existing results in the affine setting.
• Assessing Per-Sample Membership Inference Vulnerability without Retraining
  
  • Dorseuil Valentin
  • Atif Jamal
  • Cappé Olivier
  , 2026. Recent work in the privacy literature shows that sample-targeted membership inference attacks (MIAs) significantly outperform untargeted approaches by a wide margin. Motivated by this observation, we address the following question: can the privacy vulnerability of individual training points be assessed without training shadow models? We show that per-sample exposure to MIA is governed not only by a point's loss, but also by a data-dependent geometric measure. In the linear setting, we derive a closed-form decomposition of individual black-box MIA vulnerability into a population leverage score and a residual loss term, making explicit how sample-dependent geometry translates into privacy exposure. Since the final layer of most modern architectures is linear, we extend this framework to deep networks and propose a surrogate score operating on last-layer representations that requires only a single trained model and no shadow models. Empirical evaluations across diverse datasets and architectures show that our score outperforms loss and gradient-norm baselines at identifying the highest-risk points under state-of-the-art attacks, providing a computationally efficient and theoretically grounded tool for per-sample privacy risk assessment.
• Erratum for Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models
  
  • Giovangigli Vincent
  • Nabet Flore
  , 2023. An inaccurate assumption has been identified in Vincent Giovangigli, Yoann Le Calvez and Flore Nabet ``Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models'', J.~Math.~Fluid Dyn., Volume 25, 82, (2023),https://doi.org/10.1007/s00021-023-00825-4 and is corrected here.
• Optimisation topologique robuste avec microstructure incertaine pour la fabrication additive métallique
  
  • Masson Hugo
  • Denimal Goy Enora
  • Peigney Michael
  • Renson Ludovic
  , 2026, pp.8 p.. Ce travail propose de prendre en compte la variabilité de microstructure observée en fabrication additive métallique à travers un modèle matériau simple et incertain. Ce modèle est intégré à une optimisation topologique par densité, pour une minimisation de souplesse. Les incertitudes son propagées par quadrature de Gauss. La méthode réduit la variabilité de performance des formes optimisées, même dans le cas de propriétés matériaux réalistes. Des premiers résultats sur un cas industriel d’aube en 3D sont présentés.
• Métamodélisation par processus gaussiens de l'analyse vibratoire de rotors sous incertitudes et analyse de sensibilité aux paramètres
  
  • Dehillerin Erwan
  • Denimal Goy Enora
  • Sinou Jean-Jacques
  , 2026. La connaissance précise des vitesses critiques de la dynamique d’un rotor est cruciale afin de prévenir ses déformations et son endommagement. Or ces grandeurs dépendent de différents paramètres, eux-mêmes sujets à des incertitudes. Ce travail cherche ainsi à étudier les propriétés dynamiques d’un rotor dont les incertitudes portent sur les paramètres de conception. Une métamodélisation par processus gaussiens permet de comprendre la propagation des incertitudes en dimension élevée et aboutit à une étude de sensibilité qui classe et catégorise les influences des paramètres.
• 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
• 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.