arXiv subjects

Youssef Marzouk

Publications and source records attributed to Youssef Marzouk.

At least 19 recordsLinked to original sources

arXiv2026

Derivative-Informed Dimension Reduction of Stochastic Differential Equations: A Path-Space Sensitivity Approach

Uncertainty in the drift of a stochastic differential equation (SDE) induces uncertainty in statistical observables defined as expected functionals on path space. We propose a derivative-informed method for identifying low-dimensional structure in the drift-to-observable map, with the aim of improving the tractability of uncertainty quantification. This approach can be viewed as a generalization of the active subspace method to infinite-dimensional input spaces and to path-space observables of SDEs. Representing uncertainty in the drift by random perturbations in a Hilbert space, we construct a path-space sensitivity operator using first-order Fréchet derivatives of the observable with respect to the perturbation. We establish conditions under which the leading eigenfunctions of this operator correspond to directions of maximum mean squared sensitivity of the observable, thus enabling low-dimensional approximation with certified error bounds. We further introduce a martingale control variate of the Fréchet derivative that reduces error in sample-based estimators of the operator. Using a Galerkin scheme for numerical discretization, we show that reduced representations of the drift from the derivative-informed basis yield more accurate approximations of mean first hitting times of SDEs than the Karhunen-Loève expansion of the same truncation order.

math.NA↗

arXiv2026

Exploiting sparse conditional dependence in remote sensing observing systems

We develop a novel filtering algorithm for remote sensing inverse problems that exploits sparsity in the precision (inverse covariance) matrix arising from conditional independence structures in spatio-temporal satellite data. Motivated by NASA's Orbiting Carbon Observatory-2 (OCO-2), Earth Surface Mineral Dust Source Investigation (EMIT), and upcoming Explorer for Artemis Geology Lunar and Earth (EAGLE) missions, we formulate a Bayesian inference framework for retrieving atmospheric and surface parameters from hyperspectral radiance measurements. Unlike traditional methods that operate on covariances and ignore spatio-temporal dependencies, our approach models the full spatio-temporal state using sparse Gaussian graphical models, which corresponds to a block-sparse precision matrix. We derive an information filter based on this structure, enabling scalable joint inference over large spatial grids with uncertainty propagation. We demonstrate the effectiveness of the algorithm in two distinct remote sensing contexts. First, we apply the method to synthetic data configured to mimic the OCO-2 mission, focusing on retrievals of atmospheric CO$_2$, and observe a 50% reduction in CO$_2$ retrieval error compared to single-pixel methods. Then, we evaluate the algorithm using imaging spectroscopy data from the Surface Biology and Geology (SBG) High Frequency Time Series (SHIFT) campaign to retrieve surface reflectance, achieving a 20% reduction in retrieval error relative to baseline approaches.

stat.CO↗

arXiv2026

Learning smooth dynamics via Kinematic Interpolants

Inferring dynamics from real-world experimental data is inherently ambiguous, as many plausible dynamical systems can explain the same positional observations at measurement times. For higher fidelity modeling of applications ranging from single-cell RNA to vortex dynamics and even flocking of birds, scientific observables increasingly also include velocity estimates that serve to resolve this issue. However, incorporating this rich information in a scalable manner, free of simulation or unnecessary approximations, has remained a difficult challenge. In this paper, we introduce Kinematic Interpolants (KI), a novel framework that extends the seminal methodology of stochastic interpolants to infer nonlinear dynamics across multiple snapshots in phase space, i.e., observables with position and velocity measurements. Empirically, we demonstrate that across several scientific datasets, KI achieves superior reconstruction of the underlying dynamics compared to existing baselines while requiring only a fraction of the training time.

cs.LG↗

arXiv2026

Dynamical Low-Rank Filters for Data Assimilation

We propose dynamical low-rank (DLRA) filters for data-assimilation problems based on stochastic differential equations (SDEs). We first consider a Kalman filter applied to a Euler-Maruyama discretization of the SDE and observation processes, and derive a DLRA Kalman-type filter, named DLRA-JMCO, by minimizing jointly the error on the mean and covariance. We then obtain a time continuous DLRA-JMCO filter as the continuous limit of the time discrete one, and introduce a corresponding particle approximation. We also propose a strategy to enrich the DLRA-JMCO filter with a simple Gaussian process to capture the neglected dynamics in the orthogonal to the DLRA subspace. This correcting term allows the DLRA subspace evolution to be directly influenced by the observation operator. We finally propose a particle filter version of the DLRA-JMCO filter. Numerical experiments assess the performance of the proposed filters.

math.NA↗

arXiv2026

A friendly introduction to triangular transport

Decision making under uncertainty is a cross-cutting challenge in science and engineering. Most approaches to this challenge employ probabilistic representations of uncertainty. In complicated systems accessible only via data or black-box models, however, these representations are rarely known. We discuss how to characterize and manipulate such representations using triangular transport maps, which approximate any complex probability distribution as a transformation of a simple, well-understood distribution. The particular structure of triangular transport guarantees many desirable mathematical and computational properties that translate well into solving practical problems. Triangular maps are actively used for density estimation, (conditional) generative modelling, Bayesian inference, data assimilation, optimal experimental design, and related tasks. While there is ample literature on the development and theory of triangular transport methods, this manuscript provides a detailed introduction for scientists interested in employing measure transport without assuming a formal mathematical background. We build intuition for the key foundations of triangular transport, discuss many aspects of its practical implementation, and outline the frontiers of this field.

stat.CO↗

arXiv2026

Multifidelity Formulations for Triangular Transport

We develop multifidelity methods for constructing triangular transport maps from samples, when high-fidelity data are scarce but lower-fidelity data are more abundant. Using this set of multifidelity data, we approximate a triangular transport map that bijectively maps between a tractable reference density and the high-fidelity target distribution. We introduce two strategies to leverage low-fidelity data: a hierarchical approach that composes maps between adjacent fidelity levels, and a non-hierarchical method that incorporates low-fidelity information through monotonicity-preserving corrections to the map parameterization. Numerical experiments compare these strategies with single-fidelity transport and demonstrate how the proposed multifidelity approaches can improve map estimation from limited high-fidelity data. To illustrate the broader utility of the learned maps, we also deploy them in a downstream amortized simulation-based inference task. This example shows that multifidelity improvements in map estimation can translate to improved conditional sampling and uncertainty quantification when high-fidelity data are scarce.

stat.CO↗

arXiv2026

Multivariate conformal uncertainty propagation in multitask atomistic simulation: Successes and pitfalls

Machine learning has become the standard tool for the design of interatomic potentials which balance efficiency and accuracy, but uncertainty quantification remains an open problem. Multiscale simulations introduce an additional challenge: robust uncertainty quantification across scales. Even within one scale, computations are often multistage, producing a sequence of target quantities, each dependent on the previous, and each with some uncertainty. Conformal methods have emerged as a model agnostic framework for recalibrating surrogate predictions to produce sets which contain the truth at a user-specified rate. For multistage workflows, we require uncertainty calibration for multiple chemical properties and atomistic configurations, and we want to propagate uncertainty sets to downstream quantities of interest. Such propagation should capture the error cancellations which occur in many downstream targets in materials science; for instance, an approximate energy difference is often more accurate than individual energy predictions. We present the first exploration of multivariate conformal methods for chemical properties, including Bonferroni-corrected hyperrectangles, hyperellipsoidal sets based on the Mahalanobis distance, and custom loss functions within conformal risk control. Calibration is applied directly to predicted energies, atomic forces, and virial stresses, then propagated to elastic constants and vacancy formation energies employing a variety of commonly considered approximate protocols in materials modeling. We highlight the benefits of building correlation predictions into the conformal procedure, making it possible to build sets which capture near symmetries and error cancellation. We conclude with a discussion of the interplay of the employed approximate computational protocol and conformal guarantees.

physics.chem-ph↗

arXiv2026

Localized Diffusion Models

Diffusion models are state-of-the-art tools for various generative tasks. Yet training these models involves estimating high-dimensional score functions, a task that in principle suffers from the curse of dimensionality. It is therefore important to understand how low-dimensional structure in the target distribution can be exploited in these models. Here we consider locality structure, which describes certain sparse conditional dependencies among the target random variables. Given some locality structure, the score function is effectively low-dimensional, so that it can be estimated by a localized neural network with significantly reduced sample complexity. This observation motivates the localized diffusion model, where a localized score matching loss is used to train the score function within a localized hypothesis space. We prove that such localization enables diffusion models to circumvent the curse of dimensionality with dimension-independent error bounds, at the price of additional localization error. Under realistic sample size scaling, we then show both theoretically and numerically that a moderate localization radius can balance the statistical and localization errors, yielding better overall performance. Locality structure also facilitates parallel training, making localized diffusion models potentially more efficient for large-scale applications.

cs.LG↗

arXiv2026

Dynamical Low-Rank Smoothing

Computational costs often make smoothing procedures prohibitive for high-dimensional data assimilation problems. To address this challenge, we propose a dynamical low-rank approximation (DLRA) methodology for smoothing concerning frameworks based on stochastic differential equations. We extend the previously developed joint mean-and-covariance optimization (JMCO) filtering setting to derive a reduced-order smoother via the Rauch--Tung--Striebel recursion and establish the corresponding Kalman--Bucy smoothing for affine drift dynamics. The resulting algorithms retain the adaptive nature of DLRA while significantly reducing the computational time and storage of the whole smoothing procedure.

math.NA↗

arXiv2026

Efficient sampling for sparse Bayesian learning using hierarchical prior normalization

We introduce an approach for efficient Markov chain Monte Carlo (MCMC) sampling for challenging high-dimensional distributions in sparse Bayesian learning (SBL). The core innovation involves using hierarchical prior-normalizing transport maps (TMs), which are deterministic couplings that transform the sparsity-promoting SBL prior into a standard normal one. We analytically derive these prior-normalizing TMs by leveraging the product-like form of SBL priors and Knothe--Rosenblatt (KR) rearrangements. These transform the complex target posterior into a simpler reference distribution equipped with a standard normal prior that can be sampled more efficiently. Specifically, one can leverage the standard normal prior by using more efficient, structure-exploiting samplers. Our numerical experiments on various inverse problems -- including signal deblurring, inverting the non-linear inviscid Burgers equation, and recovering an impulse image -- demonstrate significant performance improvements for standard MCMC techniques.

math.NA↗

arXiv2026

Goal-oriented learning of stochastic differential equations using error bounds on path-space observables

Stochastic differential equations (SDEs), which serve as the governing equations for dynamical systems in a broad range of applications, can become cost-prohibitive for numerical simulation at scales necessary for quantifying key properties. Surrogate models of the drift function of an SDE, learned from data of the high-fidelity system, are routinely used to increase the efficiency of simulation and prediction of properties. However, standard choices of loss function for learning the surrogate model fail to provide error guarantees in certain path-dependent observables, such as transition times. This paper introduces an error bound for path-space observables and employs it as a novel variational loss for the goal-oriented learning of the drift function of a SDE. We show the error bound holds for a broad class of observables, including mean first hitting times on unbounded time domains. We derive an analytical gradient of the goal-oriented loss by leveraging the formula for Fréchet derivatives of expected path functionals, which remains tractable for implementation in stochastic gradient descent schemes. We demonstrate that surrogate models of overdamped Langevin systems developed via goal-oriented learning achieve improved accuracy in predicting the statistics of a first hitting time observable and robustness to distributional shift in the data.

stat.ME↗

arXiv2026

Quantitative Wasserstein Propagation of Chaos for Transport Ensemble Filters

We develop a general probabilistic framework for analyzing propagation of chaos in transport ensemble filters (TEFs), a broad class of interacting particle systems that are used to approximate the sequence of state distributions in hidden Markov models given a history of observations. This class of transport-based filtering algorithms includes the widely used ensemble Kalman filter (EnKF), based on affine updates at each filtering step, as well as the ensemble stochastic map filter (EnSMF), which employs nonlinear updates. For this class, we identify the limiting mean-field dynamics. We then establish non-asymptotic, high-probability, pathwise Wasserstein convergence of the interacting particle system to an i.i.d. ensemble drawn from this mean-field limit at the Monte Carlo rate. Convergence to the mean-field law itself follows with the usual dimension-dependent empirical Wasserstein rate. The proof combines a synchronous coupling construction with stability of moments and tails under conditioning, together with quantitative estimates for the propagation of the underlying dynamics through the interacting particle system. Applying our theory to both the EnKF and the EnSMF yields the first non-asymptotic, high-probability convergence guarantees for TEFs.

math.ST↗

arXiv2026

Dynestyx: A Probabilistic Programming Library for Dynamical Systems

State-space models (SSMs) are the standard formalism for Bayesian treatment of dynamical systems, with natural applications in statistics, signal processing, and machine learning. Despite their importance in both theory and application, dynamical systems have proven difficult to incorporate in modern probabilistic programming languages (PPLs), making state-of-the-art methods less accessible to practitioners and introducing friction in following the "Bayesian workflow." We introduce dynestyx, a probabilistic programming library with first-class support for SSMs, including state-of-the-art methods in the estimation of both states and parameters. Through a single, unified interface, users may specify arbitrary priors for discrete-time or continuous-time dynamical systems, perform inference over mixed-effect data, and make state and parameter estimates with principled uncertainty quantification.

stat.ML↗

arXiv2026

Optimizing Irreversible Perturbations of the Unadjusted Langevin Algorithm

Irreversible perturbations accelerate the convergence of Langevin dynamics, breaking detailed balance while preserving the invariant measure. The design of optimal irreversible perturbations has been studied in the continuous-time Gaussian setting, but extensions to non-Gaussian target distributions, and the impact of time discretization on the design of optimal perturbations, have not been well understood. Numerical discretizations of Langevin dynamics introduce bias, which is typically exacerbated by irreversible perturbations; handling this interaction demands a joint treatment of acceleration and accuracy. This paper develops a systematic framework for optimizing position-independent irreversible perturbations of the unadjusted Langevin algorithm (ULA). We formulate a constrained optimization problem that simultaneously accounts for mixing efficiency and discretization bias, where the former is characterized by a spectral gap analogue and the latter is quantified via a weighted expected squared jump distance. Within this framework, we derive an explicit characterization of the optimal position-independent irreversible perturbation. Extensive numerical experiments demonstrate that our design yields faster convergence with controlled bias, and improves mean squared estimation errors compared to other choices of irreversible perturbation.

math.NA↗

arXiv2026

Stein Kernelized Molecular Dynamics for Active Learning of Interatomic Potentials

Machine learning interatomic potentials (MLIPs) enable efficient and accurate atomistic simulations but depend critically on the quality and diversity of the training data. We introduce Stein kernelized molecular dynamics (SKMD), an enhanced sampling method that uses interacting particle dynamics to acquire informative training configurations for the active learning and fine-tuning of MLIPs. SKMD corresponds to a stochastic variant of Stein variational gradient descent that is adapted for molecular dynamics by incorporating asynchronous particle updates and a kernel of global atomic descriptors, which provides a symmetry-aware measure of configurational similarity. Unlike other enhanced samplers used in molecular dynamics, SKMD preserves the Boltzmann distribution as the asymptotic distribution of the dynamics. This property enforces a balance between the exploration of diverse configurations and attraction toward high-probability regions of the energy landscape. We further propose an approach to efficient online data acquisition using an adaptive stopping criterion that selects non-redundant training data over the course of simulation. We demonstrate SKMD for the active learning of a neural network model of the Müller-Brown potential and the fine-tuning of a MACE interatomic potential for alanine dipeptide. Compared to active learning baselines, our method achieves higher model accuracy in fewer training iterations with the same number of acquired training samples.

cs.LG↗

arXiv2026

Bayesian optimal experimental design with Wasserstein information criteria

Bayesian optimal experimental design (OED) provides a principled framework for selecting observations or experiments. We introduce new Bayesian design criteria based on the expected Wasserstein-$p$ distance between the prior and posterior distributions, termed Wasserstein information criteria. These criteria have many parallels with the widely used expected information gain (EIG) criterion, which instead relies on the Kullback--Leibler divergence. We show that the Wasserstein-$2$ criterion admits a closed-form solution in the linear-Gaussian setting, a property which can be used for more general approximation schemes, and contrast this solution with classical notions of Bayesian alphabetic optimality. Then we develop a stability analysis of the Wasserstein-$1$ criterion, wherein we bound errors induced by perturbations of the prior or likelihood. We partially extend this analysis to the Wasserstein-$2$ criterion. In particular, these results yield error rates for empirical approximations of the prior. We then illustrate the computability of the Wasserstein-$2$ criterion and demonstrate our approximation rates through simulations.

stat.ME↗

arXiv2026

Preserving linear invariants in ensemble filtering methods

Data assimilation combines dynamical models with observations to improve state estimates. Ensemble filters sequentially assimilate observations by updating a set of samples over time, alternating between a forecast and an analysis step. Accurate and robust predictions often require preserving critical invariants such as mass, stoichiometric balance of chemical species, and electrical charge. While modern numerical solvers maintain these invariants, existing invariant-preserving analysis steps are limited to Gaussian settings. Furthermore, they can be incompatible with regularization techniques such as inflation and covariance tapering. In this work, we focus on preserving linear invariants in non-Gaussian filtering problems. Leveraging tools from measure transport theory, we introduce a novel class of nonlinear ensemble filters that preserve any desired linear invariants. Notably, we recover a constrained formulation of the Kalman filter for the special case of the Gaussian setting. We also demonstrate how to combine preserving invariants with regularization techniques in the ensemble Kalman filter. Numerical experiments illustrate the benefits of preserving linear invariants in both ensemble Kalman filters and transport-based nonlinear ensemble filters.

stat.CO↗

arXiv2026

One-Shot Generative Flows: Existence and Obstructions

We study dynamic measure transport for generative modeling, focusing on transport maps that connect a source measure $P_0$ to a target measure $P_1$ by integrating a velocity field of the form $v_t(x) = \mathbb{E}[\dot X_t \mid X_t = x]$, where $X_\bullet = (X_t)_t$ is a stochastic process satisfying $(X_0,X_1)\sim{P_0}\otimes{P_1}$ and $\dot X_t$ is its time derivative. We investigate when $X_\bullet$ induces a \emph{straight-line flow}: a flow whose pointwise acceleration vanishes and is therefore exactly integrable by any first-order method. First, we develop multiple characterizations of straight-line flows in terms of PDEs involving the conditional statistics of the process. Then, we prove that straight-line flows under endpoint independence exhibit a sharp dichotomy. On the one hand, we construct explicit, computable straight-line processes for arbitrary Gaussian endpoints. On the other hand, we show that straight-line processes do not exist for targets with sufficiently well-separated modes. We demonstrate this obstruction through a sequence of increasingly general impossibility theorems that uncover a fundamental relationship between the sample-path behavior of a process with independent endpoints and the space-time geometry of this process' flow map. Taken together, these results provide a structural theory of when straight-line generative flows can, and cannot, exist.

stat.ML↗