Publications

2026

• MSAlign: Aligning Molecule and Mass Spectra representations for Metabolite Identification
  
  • Krzakala Paul
  • Melo Gabriel
  • Lançon Camille
  • Laclau Charlotte
  • Flamary Rémi
  • Thévenot Etienne
  • d'Alché-Buc Florence
  , 2026. Accurately identifying metabolites i.e. small molecules from mass spectrometry data remains a core challenge in metabolomics, with broad applications in drug discovery, environmental analysis, and clinical research. We address the Molecule Retrieval task, which consists in recovering the chemical structure of a metabolite from its MS/MS spectrum given a set of candidate molecules. We make three contributions. First, we propose a unified framework encompassing recent approaches based on representation alignment and contrastive learning. Second, we introduce MSAlign, a lightweight model that achieves state-of-the art performances by aligning two frozen foundation models (DreaMS for mass spectra and MolDeBERTa for molecules) and demonstrate that a score fusion strategy further improves the performance for a very small computational cost. Third, we investigate a long-standing evaluation problem: data splitting strategies in molecule retrieval implicitly trade off data leakage against domain shift. We formalize this tension by introducing a quantitative measure of distribution shift, and use it to evaluate splitting strategies in existing benchmarks. All datasets, splits, candidate sets, and a unified implementation of MSAlign and baselines are publicly released to support reproducible research.
• High-order adaptive discontinuous finite elements for the shallow water equations with sub-grid irregular bathymetry
  
  • Arpaia Luca
  • Orlando Giuseppe
  • Ferrarin Christian
  • Bonaventura Luca
  Advances in Water Resources, Elsevier, 2026, 217, pp.105472. We present a discontinuous finite element method for the shallow water equations which exploits high-resolution realistic bathymetry data without any regularity assumption, also in the case of high-order discretizations. We prove a number of mathematical properties specific to the proposed method that is well-balanced, mass-conserving and positivity-preserving under a mild CFL condition also in the presence of wet-dry fronts. The method includes a consistent conservative discretization for passive tracers. We use a high-order Discontinuous Galerkin (DG) method as implemented in the deal.II library. This environment provides efficient and native parallelization techniques and automatically handles non-conforming meshes to implement adaptive strategies which are tested in a coastal environment. Idealized test cases show the robustness in presence of irregular bathymetries also with under-resolved features at the grid scale. A benchmark with realistic bathymetry and a complex domain shows the potential of the proposed discretization for adaptive simulations of coastal flows. (10.1016/j.advwatres.2026.105472)
  DOI : 10.1016/j.advwatres.2026.105472
• Transport-based interpolation of dyadic meshes for problems with variable conditions
  
  • Dornier Hugo
  • Le Maitre Olivier
  • Congedo Pietro Marco
  • Niault Théotime
  , 2026. <div><p>Mesh adaptation reduces the cost of simulations with localized solution features, but repeated adaptation can become prohibitive in parametric studies. Selecting a precomputed library mesh may be insufficient when the refined region moves with the operating condition. We introduce transport-based mesh interpolation (TMI), which constructs an adapted mesh for a queried parameter by interpolating nearby meshes from an offline library. Each dyadic mesh is represented by the empirical measure of its cell centers, encoding refinement patterns through spatial point density. The interpolated representative is defined as the transport barycenter of neighboring mesh measures, using parametric barycentric weights. Geometric rules then reconstruct an admissible dyadic mesh from this representative. The proof-of-concept implementation uses a sliced Wasserstein approximation for simplicity and efficiency. Projection experiments on moving hyperbolictangent cuts isolate interpolation and reconstruction behavior, while a reaction-diffusion problem with a moving triangular source evaluates PDE-adapted meshes. In the tested configurations, the depth-informed and max-one-point reconstructions produce meshes with errors consistent with direct adaptation, with a favorable accuracy-size compromise for the depth-informed variant. Moreover, amortized online TMI becomes faster than full adaptation for the finest tested tolerances.</p></div>
• Uniform-in-time propagation of chaos for singular $L^k$ interaction kernels with rates
  
  • Cazacu Nicoleta
  • Richard Alexandre
  • Tomasevic Milica
  , 2026. Based on a semigroup approach, we prove quantitative uniform-in-time propagation of chaos for two systems of interacting particles with singular interaction kernels $K$ of type $L^k$, $k&gt;d$, and a contractive Lipschitz confinement force field. First, we consider a system with moderate interaction, which means that the particles interact through the mollified kernel $K\ast V^N$, for some approximation of identity $V^N = N^{dα} V(N^α\cdot),\, α&gt;0$. Assuming that $\|K\|_{L^k}$ is sufficiently small, we obtain that for any $\varepsilon&gt;0$ the uniform-in-time rate is of order $N^{-\min\left(α,\frac12-\frac{dα}{2}-\varepsilon\right)}$, plus an exponentially small in time contribution of the initial condition. Second, for the classical, non-mollified particle system we establish uniform-in-time propagation of chaos with rate $N^{-\min\left(α,\frac12-\frac{dα}{2}-\varepsilon\right)} + \|K-K\ast V^N\|_{L^{k}}$, plus an exponentially small in time contribution of the initial condition. The method relies on the regularity properties of the solution of the associated nonlinear Fokker-Planck equation combined with estimates of the Duhamel formulation of the mollified empirical measure, together with partial Girsanov transforms for the non-mollified system of particles.
• Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection
  
  • Raoul Gaël
  , 2026. We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to steady-states that are locally unique. The analysis is based on Wasserstein distance inequalities using a uniform lower bound on distributions. These inequalities are coupled to tail estimates to show the stability of the steady-states.
• Certified Per-Instance Unlearning Using Individual Sensitivity Bounds
  
  • Benarroch Hanna
  • Atif Jamal
  • Cappé Olivier
  , 2026. Certified machine unlearning can be achieved through noise injection, with differential privacy guarantees obtained by calibrating the noise to worst-case sensitivity. Yet, the influence of individual training points is highly heterogeneous, making such uniform calibration unnecessarily conservative for most deletion requests. We consider certified per-instance unlearning, a data-dependent notion of certified deletion in which the unlearning mechanism and its certificate can adapt to the point being removed. Our framework turns point-specific sensitivity information into formal unlearning guarantees, allowing the amount of noise to reflect the difficulty of the deletion request rather than the worst case. When training is deterministic, we derive explicit per-instance unlearning certificates. For noisy gradient descent, in the case of ridge regression, similar certificates are obtained through high-probability sensitivity bounds. Experiments on NYU-Depth with frozen DINOv2 features show substantial variation in deletion difficulty across points, and demonstrate that per-instance calibration can substantially improve utility over uniform worst-case calibration.
• Counterfactual Certification for Sequential Unlearning with Long-Run Performance Guarantees
  
  • Benarroch Hanna
  • Atif Jamal
  • Cappé Olivier
  , 2026. Certified machine unlearning ensures the proximity between the unlearned model and a model retrained without the data to be forgotten. Extending this framework to sequential deletion requests, typically arriving as groups of variable size (e.g., all records of one user or of coordinated users), is challenging. For unlearning methods based on fine-tuning on the retain sets, this leads to either degrading performance or increasing computation as the number of unlearning requests grows. For independent successive requests, we instead certify each model release against a counterfactual unlearning trajectory in which the group being deleted never took part. With a fixed fine-tuning budget per deletion, we show that, for smooth strongly convex objectives, this approach keeps the performance loss bounded, uniformly in the number of requests. Experiments with multiclass logistic regression on FEMNIST show that the resulting algorithm needs fewer corrective steps than existing sequential methods and outperforms training-time differential privacy, particularly when group sizes are heterogeneous.
• Bayesian calibration with model discrepancy of a two-group interfacial area transport model for upward bubbly flow in a square duct
  
  • Janati Idrissi Sanae
  • Rodio Maria Giovanna
  • Congedo Pietro Marco
  , 2026. Predictive simulation of bubbly flows with the two-fluid model relies on interfacial area transport equations (IATE) whose coalescence and breakup source terms involve empirical coefficients calibrated on restricted sets of experiments. This paper presents a Bayesian calibration, with explicit model discrepancy, of the two-group IATE source term coefficients embedded in the industrial CFD solver Neptune_CFD, using the upward air-water bubbly flow in a vertical square duct measured by Sun as reference data. A close examination of the experimental database reveals internally inconsistent inlet conditions, which motivates a calibration framework in which inlet boundary conditions are treated as uncertain and inferred jointly with the closure coefficients. Global sensitivity analysis shows that the wake-entrainment coefficient dominates the output variance when the inlet conditions are controlled, whereas inlet uncertainty overwhelms all parametric effects when it is present. Because each simulation is quite expensive (approximately 7.9 hours on 280 cores), calibration is made tractable by Gaussian-process surrogates constructed with an adaptive, posterior-driven design strategy that concentrates code evaluations in regions of high posterior probability and converges markedly faster than space-filling designs. Calibrations on synthetic and experimental data demonstrate that neglecting model discrepancy produces biased coefficient estimates and overconfident predictions, while explicitly modeling it separates structural model error from measurement noise and parametric uncertainty. The calibrated model substantially reduces the bias on void fraction and liquid velocity profiles and the inferred discrepancy consistently reveals a structural deficiency of the source term formulation in the near-wall region.
• Subordination of discrete snakes
  
  • Aurillard Antoine
  • Mourichoux Mathieu
  , 2026. <div><p>Motivated by applications in random geometry, we investigate the notion of subordination of snakes in the discrete setup. More precisely, given a random walk W indexed by a tree T and with steps in {..., -1, 0, 1}, we consider its subordinate tree obtained by contracting every edge of T that does not lead to a new record of the walk W. When the underlying tree T is a Bienaymé-Galton-Watson tree, we characterize the distribution of this subordinate tree. In particular, when T has a critical offspring distribution in an α-stable domain of attraction with α ∈ (1, 2], and under a light tails assumption on the steps, we prove that the associated subordinate tree is itself a Bienaymé-Galton-Watson tree with an offspring distribution in an α+1 2 -stable domain of attraction. Along the way, we obtain the asymptotic tail of the maximal displacement of the critical branching random walk W in this stable regime, under minimal assumptions. Finally, we use these results to prove scaling limit statements about the subordinate tree.</p><p>1 Bienaymé trees are also known as Galton-Watson trees. See [1] for a quick discussion on those terminologies. 1</p></div>
• Quantitative stability of optimal transport via regularization
  
  • Wang Songbo
  • Wang Xiaozhen
  , 2026. We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an $L^p$ density, $p \geqslant 1$, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the $W_2$ Hölder exponent $(p-1)/(4p-2)$. The constant depends on the dimension through three explicit source quantities: the variance, the total variation of the reference density and the $L^p$ norm of the weight. The proof uses a second-order bound on the averaged regularized Fenchel gap. By restricting the slopes of the potential, we also treat targets with a bounded moment of any order greater than two. In both estimates, the condition is imposed on one target, while the other can be any probability measure with finite second moment. For Gaussian and isotropic log-concave sources, the constants grow polynomially with the dimension. For $L^p$ changes of Gaussian measure, the factor in the bounded-target estimate is $d^{(p-1)/(4p-2)}$. For each $p$, a fixed source in two dimensions proves optimality of both stability exponents in this source class. A Gaussian construction provides polynomial lower bounds on the dimension dependence.
• Exact Asymptotics for the 2D Euclidean Random Matching Problem
  
  • Goldman Michael
  • Armegioiu Victor
  • Grotto Francesco
  • Trevisan Dario
  , 2026. <div><p>We determine the exact first-order asymptotics of the expected optimal cost in two-dimensional random bipartite matching, for every finite power cost q ≥ 1, on the flat torus. In the endpoint case q = 1, this answers a question by Talagrand, in the periodic case. The argument involves the closely related asymptotics of the energy of the solution of the p-Poisson equation with a regularized white-noise source. In the limit of vanishing regularization parameter, we identify this energy as the solution of a Variational Martingale Problem on a limiting Gaussian filtration, whose value is characterized by a parabolic Monge-Ampère flow.</p></div>
• A shape theorem for a radially excited random walk in dimensions d >= 2
  
  • Singh Arvind
  , 2026. We consider a once-excited random walk on $\mathbb{Z}^d$, $d\ge2$, where the walk on its first visit has a bias of constant strength $\beta>0$ toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape theorem: the trace after $n$ steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to $n^{1/(d+1)}$, and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.
• Systematic modelling of a large-scale, historicised, well-detailed building land use, with the fusion of massive and heterogeneous datasets
  
  • Colomb Maxime
  • Perret Julien
  PLoS ONE, Public Library of Science, 2026, 21 (10), pp.e0359346. Buildings are the primary venues for human activity. Generating precise numeric building models is of great interest for many applications. At regional and national scales, many different formalisms are employed to represent the various components of buildings, yet the building land use is frequently oversimplified — even when mixed-use buildings are considered. Inferring a realistic building land use, defined as an inventory of the distinct functional spaces hosted by a building, brings valuable knowledge for building models. Manual measurement can produce highly realistic interior models, but such approaches are labour-intensive and must be conducted on a case-by-case basis. We introduce a fully automated workflow that systematically simulates a well-described building land use by integrating three complementary tasks: (1) use-oriented building object modelling – defining building objects that capture building land use, (2) activity census and conflation – aggregating and reconciling activity records from heterogeneous datasets, and (3) logical attribute generation – deriving detailed attributes for housing units and activities in relation to the characteristics of their host building and other environmental factors. The workflow adapts to varying levels of data availability and attribute richness, and it supports “recent-period historicisation” (i.e., reconstructing past states of the built environment). Because the entire pipeline is automated, it can be deployed at a large scale; we demonstrate its application to an entire country. We evaluate the method through multiple validation procedures, including comparisons with ground truth measurements, showing that the approach reliably reproduces building land use. (10.1371/journal.pone.0359346)
  DOI : 10.1371/journal.pone.0359346
• A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent
  
  • Elbar Charles
  • Fernández-Jiménez Alejandro
  • Santambrogio Filippo
  , 2026. The Li--Yau and Aronson--Bénilan estimates are fundamental differential inequalities for heat and porous-medium type equations: they control the pressure pointwise, imply differential Harnack inequalities, and yield sharp smoothing and decay rates. We develop analogues for the critical parabolic--elliptic Keller--Segel system on $\mathbb{R}^d$, where the nonlinear diffusion competes with the aggregation on the same scale. Our main tool is the effective pressure $v=p-u$, where $p=\log\rho$ in dimension $2$ and $p=\frac{m}{m-1}\rho^{m-1}$ for $d&gt;2$, with $m=2-\frac{2}{d}$. In terms of $v$ the equation becomes $ \partial_t\rho=\operatorname{div}(\rho\nabla v), $ which yields Li--Yau estimates in dimension $2$ and Aronson--Bénilan estimates for $d&gt;2$. In each case we obtain quantitative lower bounds on $\Delta v$, showing that the classical pressure estimates survive, in a suitable form, in the presence of the nonlocal Keller--Segel interaction. On one hand these estimates recover, in a unified way, results that have so far been obtained with different techniques; on the other, they yield new ones and they have the potential to open the door to a broader program for regularity theory in the Keller--Segel system and for aggregation--diffusion equations where the pressure becomes a systematic tool. For nonnegative finite-measure initial data of sufficiently small mass (assuming only a logarithmic moment in dimension $2$), we construct global solutions that are smooth for positive times and satisfy $\|\rho_t\|_{L^\infty}\leq C/t$ in general dimension. For initial data with finite free energy, we prove global $L^\infty$ bounds of order $1+t^{-1}$ in the subcritical and critical mass regime. We introduce in this setting a new notion of critical mass, which is the minimal mass $M_c^{\mathrm{sub}}(d)$ of subsolutions of the elliptic Lane--Emden equation (Liouvile in the two-dimensional case). We then prove that this minimal mass is the expected value $8\pi$ for $d=2$ and, more generally, coincides with the the Keller--Segel critical mass coming from the HLS inequality for $d\geq 3$. This requires reducing the computation of the minimal mass to a 1D optimal control problem. Along the way we develop several tools of independent interest: $L^\infty$ bounds derived from lower estimates on $\Delta v$, critical log-HLS/HLS type estimates for families far from a Dirac mass, Hessian bounds for the Newtonian potential, and detailed Lipschitz and Hessian estimates for bounded solutions of porous-medium equations with drifts on the whole space, which are required to justify the comparison principle.
• Ancestral lineages under horizontal trait transfer: spinal decomposition and time reversal beyond stationarity
  
  • Deangeli Bravo Mateo
  , 2026. We investigate a stochastic individual-based population model with trait-dependent reproduction, death, competition, variation, and deleterious transfer. Nonlinearities appear in the competition term and the transfer form, which moreover is asymmetric. In the large-population limit, the process converges to a deterministic measure-valued equation. The latter helps us to uncover the past history of living individuals. We replace in the interactions the original stochastic process with the deterministic large population approximation obtaining a nonhomogeneous measure-valued branching process, for which we use a spinal decomposition, generalizing a method that existed only for the stationary case. Reversing time, we identify the law of typical ancestral lineages. This approach encompasses a variety of settings and requires delicate functional analysis. In contrast to earlier approaches in the literature, which relied either on stochastic calculus or on favourable jump structures under stationarity assumptions, our method provides a more direct and more general point of view.
• Continuous-state branching processes with Lévy-Khintchine drift-interaction: Laplace duality and Fellerian extensions
  
  • Foucart Clément
  • Rebotier Félix
  , 2026. 72 pages, 1 figure
• Global Energy Solutions for Normalized Linear Stochastic Schrödinger Equations
  
  • Hérouard Théo
  , 2026. In this paper, we study nonlinear Schrödinger equations arising in the study of Markovian open quantum systems. These equations are derived from the normalization of linear stochastic Schrödinger equations. First, we prove the global well-posedness of the linear equation in the energy space, which is less regular than the usual space for this equation. Then, we show the indistinguishability of solutions to the nonlinear SPDE under a relaxation of the usual monotonicity assumption. Finally, by proving the pathwise uniqueness and normalizing the solutions to the linear equation, we are able to prove the well-posedness of solutions to the required class of nonlinear stochastic equations.
• Who Drives the System? Classifying Mean-Field Particle Systems from Trajectory Data
  
  • Denis Christophe
  • Dion-Blanc Charlotte
  • Liu Yating
  , 2026. <div><p>We study the supervised classification problem for interacting particle systems (IPS) from trajectory observations. The IPS belongs to one of K classes, each characterized by a distinct interaction drift. Given a learning dataset, the goal is then to predict the class label of a newly observed system based on the trajectory of a single particle. This setting raises several statistical challenges. First, the particles within each system interact and are therefore dependent. Second, although the full particle system is Markovian, the trajectory of a single particle is not. To address these difficulties, we exploit the McKean-Vlasov limit of the IPS, which describes the dynamics of a typical particle as the number of particles tends to infinity.</p><p>We propose a plug-in classification procedure based on estimating the interaction drift associated with each class from discretely observed trajectory data. Hence, for each class, we provide a new nonparametric estimator of the drifts by minimizing a ridge-regularized least-squares contrast over a B-spline basis. In particular, our theoretical findings reveal that the convergence rate of the resulting classifier is of order N -1/6+ε for any ε &gt; 0, where N denotes the number of particles in each system. Numerical experiments illustrate the performance of the proposed method and show that it outperforms an end-to-end neural network baseline that does not exploit the underlying particle-system structure.</p></div>
• On non-uniqueness of phase retrieval for continuous and discrete Fourier transforms
  
  • Novikov Roman
  • Xu Tianli
  , 2026. We consider the problem of phase retrieval for continuous and discrete Fourier transforms in multidimensions. Using finite difference operators, we give a large class of unexpected examples of non-uniqueness for this problem, including examples with the sparsity condition. A prototype of this construction in the continuous case is given in the work Novikov, Xu (JFAA,2026), using linear differential operators. The construction of the present work also yields a large class of non-trivial Pauli partners, i.e., different functions with the same intensities in both configuration and Fourier domains. We also consider the problem of phase retrieval with background information. We show that this problem is not uniquely solvable if the known and unknown function parts are not sufficiently disjoint.
• Bayesian Calibration with Functional Outputs Using Elastic Partial Matching
  
  • Casteras Paul
  • Bect Julien
  • Garnier Josselin
  • Salin Gwenaël
  , 2026. Calibrating a simulation model involves estimating its parameters by comparing model outputs with experimental data, so that simulation results faithfully reproduce the experimental observations. When the outputs are functions of time, there are multiple ways to quantify the discrepancy between experimental and simulated curves. A recent approach based on elastic functional data analysis decomposes a functional output into two components: a function temporally aligned to a template, and the corresponding warping function. This decomposition splits the problem into two independent calibration tasks, thereby addressing functional misalignment. However, it assumes that experimental and simulated curves share the same temporal support, an assumption often violated in practice when initial or end times are themselves uncertain or depend on the calibration parameters. In this work, we reinterpret the decomposition step as an approximation to a more general Bayesian calibration problem that incorporates an error term on the time axis. This perspective allows us to naturally extend the framework to a broader family of time warpings with varying initial or end times, using partial elastic alignment. We illustrate the method on a synthetic test case, comparing it with existing Bayesian calibration methods and demonstrating improved surrogate performance and error modeling. We then apply the proposed approach to the calibration of an equation of state (a thermodynamic equation relating the state variables of a material).
• Introduction to the Special Issue on Benchmarking in Multi-Criteria Optimization
  
  • Brockhoff Dimo
  • Naujoks Boris
  • Purshouse Robin
  • Tušar Tea
  ACM Transactions on Evolutionary Learning and Optimization, ACM, 2026, 6 (3), pp.1-2. (10.1145/3839244)
  DOI : 10.1145/3839244
• Kernel Density Regression Estimation with Covariates on Riemannian Manifolds
  
  • Rihab Gharbi
  • Khardani Salah
  • Abdillahi Isman Mohamed
  • Yao Anne-Françoise
  , 2026.
• Invasion dynamics with vanishing fitness for a quasi-critical birth-death process
  
  • Bansaye Vincent
  • Belmabrouk Nadia
  • Erny Xavier
  • Girel Simon
  , 2026. We study the invasion dynamics of populations exhibiting positive density-dependent effects. We start with a single individual and consider a single-type birth and death process. The initial individual growth rate vanishes but it increases with the population density, proportionally to the number of individuals divided by a scaling parameter $K$. Before reaching the macroscopic scale~$K$, the population process is almost critical. We prove that the probability for the population to reach the macroscopic level $K$ decreases as $1/\sqrt{K}$ as $K$ goes to infinity. We also describe the associated trajectories and show that invasion can be split into three time periods. First, the process needs to escape from zero, and conditioning on survival, it grows linearly until the order $\sqrt{K}$. The scaled process is approximated by a diffusion, as for critical branching process, with an additional drift term coming from cooperation, which breaks the branching property. Second, in intermediate scale $\sqrt{K}$, we observe another diffusion, surviving with positive probability, without conditioning. Finally, beyond $\sqrt{K}$ scale, the process can be approximated by a classical macroscopic ODE limit. The proof of the first phase involves change of probability and characterization of uniform integrability of martingales, while the two other phases rely on uniform approximations on polynomial time scales.
• CONVERGENCE ANALYSIS OF A DISCRETE DUALITY FINITE-VOLUME METHOD FOR CAHN-HILLIARD EQUATIONS WITH SURFACTANTS
  
  • Castellano Marguerita
  • Goudenège Ludovic
  • Nabet Flore
  , 2026. We propose a Discrete Duality Finite Volume (DDFV) scheme for a phase-field model of surfactant dynamics during the phase separation of a two-phase flow. The model couples two Cahn-Hilliard equations, for the phase-field variable and for the surfactant concentration. We discretize the model in space via the DDFV method, well-suited to general meshes and nonlinear diffusion, and in time via a semi-implicit Euler scheme. The discrete scheme and the corresponding numerical analysis rely on two dedicated operators discretizing the nonlinear term coupling the surfactant concentration and the phase-field gradient consistently across the primal, dual and diamond meshes. We first prove the existence of a discrete solution at each time step, via a topological degree argument. We then prove that, as the mesh size and time step tend to zero, the approximate solutions converge to a weak solution of the continuous problem. The existence of a solution to the continuous problem then follows as a consequence. To our knowledge, the finite volume analysis, the convergence of the scheme and the existence of a continuous solution are all new for this model. We finally report numerical simulations illustrating the interface-driven surfactant dynamics, including a benchmark showing the stabilization of the interfaces under high surfactant concentration.
• Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization
  
  • Bonet Clément
  • Aubin-Frankowski Pierre-Cyril
  • Mroueh Youssef
  , 2026. Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.