arXiv subjects

Andrea Zanoni

Publications and source records attributed to Andrea Zanoni.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic modeling of Fourier modes in two-dimensional turbulence via filtered white noise

Modeling turbulent flows by a random Fourier decomposition is a classical procedure in order to use simplified models of turbulence in heat transport and other applications. We investigate the Fourier time series of two-dimensional Navier-Stokes equations with friction and damping, forced at intermediate scales, and identify significant statistical structures. In particular, we find the existence of a typical time correlation length, and propose a stochastic model for the Fourier components. Finally, we compute the transport of a passive scalar under advection-diffusion dynamics by means of direct numerical simulation of the stochastic damped Navier-Stokes equation and compare it with analytical predictions of the effective diffusion produced by the stochastic model.

math-ph↗

arXiv2026

Enabling stratified sampling in high dimensions via nonlinear dimensionality reduction

We consider the problem of propagating the uncertainty from a possibly large number of random inputs through a computationally expensive model. Stratified sampling is a well-known variance reduction strategy, but its application, thus far, has focused on models with a limited number of inputs due to the challenges of creating uniform partitions in high dimensions. To overcome these challenges, we propose a simple methodology for constructing an effective stratification of the input domain that is adapted to the model response. Our approach leverages neural active manifolds, a recently introduced nonlinear dimensionality reduction technique based on neural networks that identifies a one-dimensional manifold capturing most of the model variability. The resulting one-dimensional latent space is mapped to the unit interval, where stratification is performed with respect to the uniform distribution. The corresponding strata in the original input space are then recovered through the neural active manifold, generating partitions that tend to follow the level sets of the model. We show that our approach is effective in high dimensions and can be used to further reduce the variance of multifidelity Monte Carlo estimators.

math.NA↗

arXiv2025

On the performance of multi-fidelity and reduced-dimensional neural emulators for inference of physiological boundary conditions

Solving inverse problems in cardiovascular modeling is particularly challenging due to the high computational cost of running high-fidelity simulations. In this work, we focus on Bayesian parameter estimation and explore different methods to reduce the computational cost of sampling from the posterior distribution by leveraging low-fidelity approximations. A common approach is to construct a surrogate model for the high-fidelity simulation itself. Another is to build a surrogate for the discrepancy between high- and low-fidelity models. This discrepancy, which is often easier to approximate, is modeled with either a fully connected neural network or a nonlinear dimensionality reduction technique that enables surrogate construction in a lower-dimensional space. A third possible approach is to treat the discrepancy between the high-fidelity and surrogate models as random noise and estimate its distribution using normalizing flows. This allows us to incorporate the approximation error into the Bayesian inverse problem by modifying the likelihood function. We validate five different methods which are variations of the above on analytical test cases by comparing them to posterior distributions derived solely from high-fidelity models, assessing both accuracy and computational cost. Finally, we demonstrate our approaches on two cardiovascular examples of increasing complexity: a lumped-parameter Windkessel model and a patient-specific three-dimensional anatomy.

stat.ML↗

arXiv2025

Neural active manifolds: nonlinear dimensionality reduction for uncertainty quantification

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, through the aid of a simultaneously learnt surrogate model with inputs on this manifold. Our method only relies on model evaluations and does not require the knowledge of gradients. The proposed dimensionality reduction framework can then be applied to assist outer loop many-query tasks in scientific computing, like sensitivity analysis and multifidelity uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy and highlight its advantages with respect to existing approaches in the literature.

math.NA↗

arXiv2025

A Fourier-based inference method for learning interaction kernels in particle systems

We consider the problem of inferring the interaction kernel of stochastic interacting particle systems from observations of a single particle. We adopt a semi-parametric approach and represent the interaction kernel in terms of a generalized Fourier series. The basis functions in this expansion are tailored to the problem at hand and are chosen to be orthogonal polynomials with respect to the invariant measure of the mean-field dynamics. The generalized Fourier coefficients are obtained as the solution of an appropriate linear system whose coefficients depend on the moments of the invariant measure, and which are approximated from the particle trajectory that we observe. We quantify the approximation error in the Lebesgue space weighted by the invariant measure and study the asymptotic properties of the estimator in the joint limit as the observation interval and the number of particles tend to infinity, i.e. the joint large time-mean field limit. We also explore the regime where an increasing number of generalized Fourier coefficients is needed to represent the interaction kernel. Our theoretical results are supported by extensive numerical simulations.

math.ST↗

arXiv2025

Nonparametric estimation of homogenized invariant measures from multiscale data via Hermite expansion

We consider the problem of density estimation in the context of multiscale Langevin diffusion processes, where a single-scale homogenized surrogate model can be derived. In particular, our aim is to learn the density of the invariant measure of the homogenized dynamics from a continuous-time trajectory generated by the full multiscale system. We propose a spectral method based on a truncated Fourier expansion with Hermite functions as orthonormal basis. The Fourier coefficients are computed directly from the data owing to the ergodic theorem. We prove that the resulting density estimator is robust and converges to the invariant density of the homogenized model as the scale separation parameter vanishes, provided the time horizon and the number of Fourier modes are suitably chosen in relation to the multiscale parameter. The accuracy and reliability of this methodology is further demonstrated through a series of numerical experiments.

math.NA↗

arXiv2025

Linearization of ergodic McKean SDEs and applications

In this article, we consider McKean stochastic differential equations, as well as their corresponding McKean-Vlasov partial differential equations, which admit a unique stationary state, and we study the linearized Itô diffusion process that is obtained by replacing the law of the process in the convolution term with the unique invariant measure. We show that the law of the nonlinear McKean process converges to the law of this linearized process exponentially fast in time, both in relative entropy and in Wasserstein distance. We study the problem in both the whole space and the torus. We then show how we can employ the resulting linear (in the sense of McKean) Markov process to analyze properties of the original nonlinear and nonlocal dynamics that depend on their long-time behavior. In particular, we propose a linearized maximum likelihood estimator for the nonlinear process which is asymptotically unbiased, and we study the joint diffusive-mean field limit of the underlying interacting particle system.

math.PR↗

arXiv2024

Filtered data based estimators for stochastic processes driven by colored noise

We consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, which we model as a sequence of Gaussian stationary processes with decreasing correlation time. We aim to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose to modify them by including filtered data. We provide a convergence analysis for our estimators showing their asymptotic unbiasedness in a general setting and asymptotic normality under a simplified scenario.

math.NA↗

arXiv2024

Stochastic gradient descent in continuous time for drift identification in multiscale diffusions

We consider the setting of multiscale overdamped Langevin stochastic differential equations, and study the problem of learning the drift function of the homogenized dynamics from continuous-time observations of the multiscale system. We decompose the drift term in a truncated series of basis functions, and employ the stochastic gradient descent in continuous time to infer the coefficients of the expansion. Due to the incompatibility between the multiscale data and the homogenized model, the estimator alone is not able to reconstruct the exact drift. We therefore propose to filter the original trajectory through appropriate kernels and include filtered data in the stochastic differential equation for the estimator, which indeed solves the misspecification issue. Several numerical experiments highlight the accuracy of our approach. Moreover, we show theoretically in a simplified framework the asymptotic unbiasedness of our estimator in the limit of infinite data and when the multiscale parameter describing the fastest scale vanishes.

math.NA↗

arXiv2024

Personalized and uncertainty-aware coronary hemodynamics simulations: From Bayesian estimation to improved multi-fidelity uncertainty quantification

Simulations of coronary hemodynamics have improved non-invasive clinical risk stratification and treatment outcomes for coronary artery disease, compared to relying on anatomical imaging alone. However, simulations typically use empirical approaches to distribute total coronary flow amongst the arteries in the coronary tree. This ignores patient variability, the presence of disease, and other clinical factors. Further, uncertainty in the clinical data often remains unaccounted for in the modeling pipeline. We present an end-to-end uncertainty-aware pipeline to (1) personalize coronary flow simulations by incorporating branch-specific coronary flows as well as cardiac function; and (2) predict clinical and biomechanical quantities of interest with improved precision, while accounting for uncertainty in the clinical data. We assimilate patient-specific measurements of myocardial blood flow from CT myocardial perfusion imaging to estimate branch-specific coronary flows. We use adaptive Markov Chain Monte Carlo sampling to estimate the joint posterior distributions of model parameters with simulated noise in the clinical data. Additionally, we determine the posterior predictive distribution for relevant quantities of interest using a new approach combining multi-fidelity Monte Carlo estimation with non-linear, data-driven dimensionality reduction. Our framework recapitulates clinically measured cardiac function as well as branch-specific coronary flows under measurement uncertainty. We substantially shrink the confidence intervals for estimated quantities of interest compared to single-fidelity and state-of-the-art multi-fidelity Monte Carlo methods. This is especially true for quantities that showed limited correlation between the low- and high-fidelity model predictions. Moreover, the proposed estimators are significantly cheaper to compute for a specified confidence level or variance.

physics.flu-dyn↗

arXiv2024

Improved multifidelity Monte Carlo estimators based on normalizing flows and dimensionality reduction techniques

We study the problem of multifidelity uncertainty propagation for computationally expensive models. In particular, we consider the general setting where the high-fidelity and low-fidelity models have a dissimilar parameterization both in terms of number of random inputs and their probability distributions, which can be either known in closed form or provided through samples. We derive novel multifidelity Monte Carlo estimators which rely on a shared subspace between the high-fidelity and low-fidelity models where the parameters follow the same probability distribution, i.e., a standard Gaussian. We build the shared space employing normalizing flows to map different probability distributions into a common one, together with linear and nonlinear dimensionality reduction techniques, active subspaces and autoencoders, respectively, which capture the subspaces where the models vary the most. We then compose the existing low-fidelity model with these transformations and construct modified models with an increased correlation with the high-fidelity model, which therefore yield multifidelity Monte Carlo estimators with reduced variance. A series of numerical experiments illustrate the properties and advantages of our approaches.

stat.ME↗

arXiv2024

Laser-written micro-channel atomic magnetometer

We demonstrate a sensitive optically-pumped magnetometer using rubidium vapor and 0.75 amg of nitrogen buffer gas in a sub-mm-width sensing channel excavated by femtosecond laser writing followed by chemical etching. The channel is buried less than 1 mm below the surface of its fused silica host material, which also includes reservoir chambers and micro-strainer connections, to preserve a clean optical environment. Using a zero-field-resonance magnetometry strategy and a sensing volume of 2.25 mm$^3$, we demonstrate a sensitivity of $\approx$ 1 $\mathrm{pT}/\sqrt{\mathrm{Hz}}$ at $10$ Hz. The device can be integrated with photonic structures and microfluidic channels with 3D versatility. Its sensitivity, bandwidth and stand-off distance will enable detection of localized fields from magnetic nanoparticles and \mul NMR samples.

quant-ph↗

arXiv2024

Eigenfunction martingale estimators for interacting particle systems and their mean field limit

We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.

math.NA↗

arXiv2024

Homogenization results for the generator of multiscale Langevin dynamics in weighted Sobolev spaces

We study the homogenization of the Poisson equation with reaction term and of the eigenvalue problem associated to the generator of multiscale Langevin dynamics. Our analysis extends the theory of two-scale convergence to the case of weighted Sobolev spaces in unbounded domains. We provide convergence results for the solution of the multiscale problems above to their homogenized surrogate. A series of numerical examples corroborate our analysis.

math.AP↗

arXiv2024

A method of moments estimator for interacting particle systems and their mean field limit

We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.

math.NA↗

arXiv2023

A Probabilistic Neural Twin for Treatment Planning in Peripheral Pulmonary Artery Stenosis

The substantial computational cost of high-fidelity models in numerical hemodynamics has, so far, relegated their use mainly to offline treatment planning. New breakthroughs in data-driven architectures and optimization techniques for fast surrogate modeling provide an exciting opportunity to overcome these limitations, enabling the use of such technology for time-critical decisions. We discuss an application to the repair of multiple stenosis in peripheral pulmonary artery disease through either transcatheter pulmonary artery rehabilitation or surgery, where it is of interest to achieve desired pressures and flows at specific locations in the pulmonary artery tree, while minimizing the risk for the patient. Since different degrees of success can be achieved in practice during treatment, we formulate the problem in probability, and solve it through a sample-based approach. We propose a new offline-online pipeline for probabilsitic real-time treatment planning which combines offline assimilation of boundary conditions, model reduction, and training dataset generation with online estimation of marginal probabilities, possibly conditioned on the degree of augmentation observed in already repaired lesions. Moreover, we propose a new approach for the parametrization of arbitrarily shaped vascular repairs through iterative corrections of a zero-dimensional approximant. We demonstrate this pipeline for a diseased model of the pulmonary artery tree available through the Vascular Model Repository.

physics.med-ph↗

arXiv2022

Laser-written vapor cells for chip-scale atomic sensing and spectroscopy

We report the fabrication of alkali-metal vapor cells using femtosecond laser machining. This laser-written vapor-cell (LWVC) technology allows arbitrarily-shaped 3D interior volumes and has potential for integration with photonic structures and optical components. We use non-evaporable getters both to dispense rubidium and to absorb buffer gas. This enables us to produce cells with sub-atmospheric buffer gas pressures without vacuum apparatus. We demonstrate sub-Doppler saturated absorption spectroscopy and single beam optical magnetometry with a single LWVC. The LWVC technology may find application in miniaturized atomic quantum sensors and frequency references.

physics.optics↗

arXiv2022

Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions

We propose a novel method for drift estimation of multiscale diffusion processes when a sequence of discrete observations is given. For the Langevin dynamics in a two-scale potential, our approach relies on the eigenvalues and the eigenfunctions of the homogenized dynamics. Our first estimator is derived from a martingale estimating function of the generator of the homogenized diffusion process. However, the unbiasedness of the estimator depends on the rate with which the observations are sampled. We therefore introduce a second estimator which relies also on filtering the data and we prove that it is asymptotically unbiased independently of the sampling rate. A series of numerical experiments illustrate the reliability and efficiency of our different estimators.

math.NA↗