Publications

2026

• 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>
• Asymptotic Stability of Equilibrium States for Cohesive Fluids
  
  • Giovangigli Vincent
  , 2026. We investigate asymptotic stability of constant equilibrium states for compressible nonisothermal cohesive fluids also termed capillary fluids or diffuse interface fluids. The density gradient is added as an extra variable and the augmented system of equations is recast into a normal form with symmetric transport first order terms, symmetric dissipative second order terms and antisymmetric cohesive second order terms. Global existence and asymptotic stability of constant equilibrium states are established by using new dissipative conditions for such augmented hyperbolic-parabolic-dispersive systems of equations. Decay estimates are obtained in all spatial dimensions by using the augmented formulation as well as estimates in Fourier spaces.
• Efficient Monte Carlo sampling of metastable systems using nonlocal collective variable updates
  
  • Schönle Christoph
  • Carbone Davide
  • Gabrié Marylou
  • Lelièvre Tony
  • Stoltz Gabriel
  The Journal of Chemical Physics, American Institute of Physics, 2026, 164 (15), pp.154107. Monte Carlo simulations are widely used to simulate complex molecular systems, but standard approaches suffer from metastability. Lately, the use of nonlocal proposal updates in a collective-variable (CV) space has been proposed in several works. Here, we generalize these approaches and explicitly spell out an algorithm for nonlinear CVs and underdamped Langevin dynamics. We prove reversibility of the resulting scheme and demonstrate its performance on several numerical examples, observing a substantial performance increase compared to methods based on overdamped Langevin dynamics as considered previously. Advances in generative machine-learning-based proposal samplers now enable efficient sampling in CV spaces of intermediate dimensionality—tens to hundreds of variables—and our results extend their applicability toward more realistic molecular systems. (10.1063/5.0318016)
  DOI : 10.1063/5.0318016
• Growing random planar network with oriented branching and fusion
  
  • Bansaye Vincent
  • Raoul Gaël
  • Tomasevic Milica
  , 2026. We consider a growing planar network where a tip grows at constant speed, branches at constant rate and inactivates when it meets a branch already created. We only consider here orthogonal branching occurring always in the same direction. This yields a spatial branching property to the growing network. The connected components of the network then form a branching process of rectangles with double immigration. Using a spine approach for a typical rectangle and coupling arguments, the study is boiled down to a one dimensional stick breaking model with aging. We can then prove long time convergence of empirical measure of the family of rectangles after polynomial rescaling. The limiting distribution and speed of convergence can be explicitly described. The proofs also rely on the description of common ancestor of rectangles in the branching structure with double immigration.
• 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.
• 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.
• Accelerating Nash learning from human feedback via Mirror Prox
  
  • Tiapkin Daniil
  • Calandriello Daniele
  • Belomestny Denis
  • Moulines Eric
  • Naumov Alexey
  • Rasul Kashif
  • Valko Michal
  • Ménard Pierre
  , 2025.
• Semi-discrete convergence analysis of a numerical method for waves in nearly-incompressible media with spectral finite elements
  
  • Ramiche Zineb
  • Imperiale Sébastien
  , 2026. In this work, we present a semi-discrete convergence analysis of a high-order space discretization approach for the computation of elastic field propagation in a nearly incompressible medium. Our approach relies on the use of high-order continuous spectral finite elements with mass-lumping. We present an approach that is valid for full hexahedral and quadrilateral meshes, where the elastic field is sought in the space of Q_k continuous finite elements and the pressure in Q_k-2 discontinuous finite elements. We further provide a proof of the inf-sup stability of the finite element discretization. This allows us to carry out error estimates for the semi-discrete problem in space, accounting in particular for quadrature errors.
• Autoregressive Multiplier Bootstrap for In-situ Error Estimation and Quality Monitoring of Finite Time Averages in Turbulent Flow Simulations
  
  • Papagiannis Christos
  • Balarac Guillaume
  • Congedo Pietro Marco
  • Le Maître Olivier P
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2026, 452, pp.118664. In Computational Fluid Dynamics (CFD), and particularly within Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES), the computational cost is largely dictated by the effort required to obtain statistically converged quantities such as time-averaged fields and higher-order moments. Despite the importance of accurately quantifying statistical uncertainty in unsteady simulations, no continuous and cost-effective, on-line method currently exists for monitoring the convergence quality of such statistics during runtime. This work introduces a novel, fully on-line bootstrapping approach to estimate the variance of finite-time averages without requiring the estimation of the flows Auto-Correlation Function (ACF). Unlike existing methods that rely on ACF estimation, which are often impractical due to excessive storage demands in large-scale simulations, or require off-line processing or a priori modeling assumptions, our method operates entirely during the simulation and incurs minimal overhead. The proposed technique employs a recursive update of bootstrap replicates of the time average, using correlated random weights generated via an autoregressive model. This formulation is computationally efficient: the update cost scales linearly with the number of bootstrap replicates and the dimensionality of the flow field, and the autoregressive model is inexpensive to evaluate. The method only requires storage of a small number of fields, making it suitable for large-scale CFD applications. We demonstrate the effectiveness of the approach on synthetic data from the Ornstein-Uhlenbeck process and on two canonical LES cases: a turbulent pipe flow and a round jet. We further discuss the methods applicability to simulations with non-uniform time stepping, highlighting its flexibility and robustness. (10.1016/j.cma.2025.118664)
  DOI : 10.1016/j.cma.2025.118664
• A stability result on optimal Skorokhod embedding
  
  • Guo Gaoyue
  , 2017. Motivated by the model- independent pricing of derivatives calibrated to the real market, we consider an optimization problem similar to the optimal Skorokhod embedding problem, where the embedded Brownian motion needs only to reproduce a finite number of prices of Vanilla options. We derive in this paper the corresponding dualities and the geometric characterization of optimizers. Then we show a stability result, i.e. when more and more Vanilla options are given, the optimization problem converges to an optimal Skorokhod embedding problem, which constitutes the basis of the numerical computation in practice. In addition, by means of different metrics on the space of probability measures, a convergence rate analysis is provided under suitable conditions.
• Constructive existence proofs and stability of stationary solutions to parabolic PDEs using Gegenbauer polynomials
  
  • Breden Maxime
  • Cadiot Matthieu
  • Zurek Antoine
  , 2026. In this paper, we present a computer-assisted framework for constructive proofs of existence for stationary solutions to one-dimensional parabolic PDEs and the rigorous determination of their linear stability. By expanding solutions in Gegenbauer polynomials, we first develop a general approach for boundary value problems (BVPs), corresponding to the stationary part of the PDE. This yields a computationally efficient sparse structure for both differential and multiplication operators. By deriving sharp, explicit and quantitative estimates for the inverse of differential operators, we implement a Newton-Kantorovich approach. Specifically, given a numerical approximation <span>u&#772;</span>, we prove the existence of a true stationary solution u within a small, rigorously quantified neighborhood of <span>u&#772;</span>. A key advantage of this approach is that the sharp control over the defect u -<span>u&#772;</span>, integrated with the spectral properties of the Gegenbauer basis, enables an accurate enclosure of the linearization's spectrum around u. This allows for a definitive conclusion regarding the (in)stability of the verified solution, which is the main contribution of the paper. We demonstrate the efficacy of this method through several applications, capturing both stable and unstable equilibrium states.
• Invasion dynamics for a quasi-critical birth-death process
  
  • Bansaye Vincent
  • Belmabrouk Nadia
  • Erny Xavier
  • Girel Simon
  , 2026. We study the dynamics of invasion of populations which enjoy positive density dependent effect. We start with a single individual and consider a single type birth and death process. The initial individual growth rate is zero and it increases. We prove that the probability that the population reaches macroscopic levels decreases as $\sqrt{K}$, where $K$ is the scaling parameter. We also describe the associated trajectories and show that invasion can be split into three time periods. First, the process needs to escape the neighborhood of $0$, and conditioning on survival, it grows linearly until the order $\sqrt{K}$. The renormalized process is approximated by a diffusion, as for critical branching process, with an additional drift term coming from cooperation. Second, in intermediate scale $\sqrt{K}$, we observe another diffusion, without conditioning. Finally, this diffusion can be approximated by classical fluid limit corresponding to macroscopic approximation by an ODE. The proof of the first phase involves change of probability and characterization of uniform integrability of martingales, while the two other phases need uniform approximations on polynomial time scales.
• Systematic modelling of a large-scaled, historicised, well-detailed building land-use, with the fusion of massive and heterogeneous datasets
  
  • Colomb Maxime
  • Perret Julien
  , 2026. 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 carried out case by case. We introduce a fully automated workflow that systematically simulates a well-described building land-use by integrating three complementary tasks: (1) usage-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 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.
• Young's law for a nonlocal isoperimetric model of charged capillarity droplets
  
  • Goldman Michael
  • Novaga Matteo
  • Prade Adriano
  , 2026. We study a variational problem modeling equilibrium configurations of charged liquid droplets resting on a surface under a convexity constraint. In the two-dimensional case with Coulomb interactions, we establish the validity of Young's law for the contact angle for small enough charges.
• Stability analysis and long-time convergence of a partial differential equation model of two-phase ageing
  
  • Breuil Luce
  , 2026. Recent biological evidence suggests the presence of a two-phase ageing process in several species. We introduce a system of two age-structured partial differential equations (PDE) representing two phases of ageing of a wild population. The model includes a coupling of both equations through birth and transition between phases and non-linearities due to competition. We show the existence, positivity and uniqueness of weak solutions in a general setting. For a simplified system of ordinary differential equations (ODE), we show existence and uniqueness of a strictly positive steady state attracting all trajectories. We study another simplification, a coupled PDE-ODE model, for which we prove existence, uniqueness and local asymptotic stability of a strictly positive steady state. Under further assumptions, but without assuming weak non-linearities, we show the global asymptotic stability of that steady state. The uniqueness of steady states and absence of oscillations in these systems show that the proportion of individuals in each phase at equilibrium is a unique feature of the model. This paves the way to ecological applications as the experimental measure of such a proportion could help gain some insight on the health of a wild population.
• An all-topology two-fluid model for two-phase flows derived through Hamilton's Stationary Action Principle
  
  • Haegeman Ward
  • Orlando Giuseppe
  • Kokh Samuel
  • Massot Marc
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, 2026, 482 (2334), pp.20250835. We present a novel multi-fluid model for compressible two-phase flows. The model is derived through a newly developed Stationary Action Principle framework. It is fully closed and introduces a new interfacial quantity, the interfacial work. The closures for the interfacial quantities are provided by the variational principle. They are physically sound and well-defined for all types of flow topologies. The model is shown to be hyperbolic, symmetrizable, and admits an entropy conservation law. Its non-conservative products yield uniquely defined jump conditions which are provided. As such, it allows for the proper treatment of weak solutions. In the multi-dimensional setting, the model presents lift forces which are discussed. The model constitutes a sound basis for future numerical simulations. (10.1098/rspa.2025.0835)
  DOI : 10.1098/rspa.2025.0835
• Nonnegativity certificates for finite semi-algebraic sets
  
  • Bender Matías R
  • Tsigaridas Elias
  • Zenkovich Alexander
  , 2026. <div><p>We introduce new certificates for nonnegativity of multivariate polynomials with rational coefficients over zero-dimensional semi-algebraic sets. They are perfectly complete, certifying every nonnegative polynomial, and perfectly sound, correctly identifying negativity. We rely on resultants computations and Rational Univariate Representations (RUR) and make no assumptions on the input.</p><p>For the univariate case, we introduce a perturbation technique that avoids root approximation and does not alter the (bit)size of the input. For multivariate polynomials, we make a reduction to the univariate case using RUR.</p><p>For the dense case, we compute a certificate in OB(d 4n+3 (d + τ )) bit operations; it involves numbers of bitsize O(d 3n+2 (d + τ )), where n is the number of variables, d the degree, and τ the maximum coefficient bitsize of the polynomials. For the sparse case, we provide the first sparse certificate based on the Newton polytope Q of the input polynomials. We compute in OB(vol(Q) 8 (n!) 8 2 5n+3 (n + τ )),</p><p>For semi-algebraic sets with s inequalities, we present two approaches. The first performs a reduction to the algebraic case and has complexity OB(2 (3ω+3)s d 5n τ ); it is purely algebraic approach and does not require root approximation. The second exploits approximate Lagrange interpolation and matches the O(sd 4n τ ) bitsize bounds of recent work by Baldi, Krick, and Mourrain [2] while improving complexity by orders of magnitude and removing all structural assumptions on the input.</p><p>Additionally, we provide a witness of negativity, ensuring that we either obtain a certificate or it does not exist.</p></div>
• Modeling the risks within the protocol Aave, with an application to portfolio allocation
  
  • Gobet Emmanuel
  • Latournerie Louis
  , 2026. Decentralized Finance (DeFi) lending and borrowing protocols enable investors to take leveraged long and short positions on digital assets without centralized intermediaries, but expose them to a distinctive form of risk: on-chain liquidation triggered by debt and collateral value fluctuations. In this work, we provide a detailed formalization of Aave's lending, borrowing, and liquidation mechanisms, grounded in the protocol's open-source implementation. In doing so, we propose a mathematical modeling of the risk of liquidation, including some stochastic approximations with the purpose of efficient analysis, with different applications. Among them, portfolio optimization problem.
• Fast, faithful and photorealistic diffusion-based image super-resolution with enhanced Flow Map models
  
  • Noble Maxence
  • Quintana Gonzalo Iñaki
  • Aubin Benjamin
  • Chadebec Clément
  , 2026. Diffusion-based image super-resolution (SR) has recently attracted significant attention by leveraging the expressive power of large pre-trained text-to-image diffusion models (DMs). A central practical challenge is resolving the trade-off between reconstruction faithfulness and photorealism. To address inference efficiency, many recent works have explored knowledge distillation strategies specifically tailored to SR, enabling one-step diffusion-based approaches. However, these teacher-student formulations are inherently constrained by information compression, which can degrade perceptual cues such as lifelike textures and depth of field, even with high overall perceptual quality. In parallel, self-distillation DMs, known as Flow Map models, have emerged as a promising alternative for image generation tasks, enabling fast inference while preserving the expressivity and training stability of standard DMs. Building on these developments, we propose FlowMapSR, a novel diffusion-based framework for image super-resolution explicitly designed for efficient inference. Beyond adapting Flow Map models to SR, we introduce two complementary enhancements: (i) positive-negative prompting guidance, based on a generalization of classifier free-guidance paradigm to Flow Map models, and (ii) adversarial fine-tuning using Low-Rank Adaptation (LoRA). Among the considered Flow Map formulations (Eulerian, Lagrangian, and Shortcut), we find that the Shortcut variant consistently achieves the best performance when combined with these enhancements. Extensive experiments show that FlowMapSR achieves a better balance between reconstruction faithfulness and photorealism than recent state-of-the-art methods for both x4 and x8 upscaling, while maintaining competitive inference time. Notably, a single model is used for both upscaling factors, without any scale-specific conditioning or degradation-guided mechanisms.
• Diffusion-based Annealed Boltzmann Generators : benefits, pitfalls and hopes
  
  • Grenioux Louis
  • Noble Maxence
  , 2026. Sampling configurations at thermodynamic equilibrium is a central challenge in statistical physics. Boltzmann Generators (BGs) tackle it by combining a generative model with a Monte Carlo (MC) correction step to obtain asymptotically unbiased samples from an unnormalized target. Most current BGs use classic MC mechanisms such as importance sampling, which both require tractable likelihoods from the backbone model and scale poorly in high-dimensional, multi-modal targets. We study BGs built on annealed Monte Carlo (aMC), which is designed to overcome these limitations by bridging a simple reference to the target through a sequence of intermediate densities. Diffusion models (DMs) are powerful generative models and have already been incorporated into aMC-based recalibration schemes via the diffusion-induced density path, making them appealing backbones for aMC-BGs. We provide an empirical meta-analysis of DM-based aMC-BGs on controlled multi-modal Gaussian mixtures (varying mode separation, number of modes, and dimension), explicitly disentangling inference effects from learning effects by comparing (i) a perfectly learned DM and (ii) a DM trained from data. Even with a perfect DM, standard integrations using only first-order stochastic denoising kernels fail systematically, whereas second-order denoising kernels can substantially improve performance when covariance information is available. We further propose a deterministic aMC integration based on first-order transport maps derived from DMs, which outperforms the stochastic first-order variant at higher computational cost. Finally, in the learned-DM setting, all DM-aMC variants struggle to produce accurate BGs; we trace the main bottleneck to inaccurate DM log-density estimation.