arXiv subjects

Bruno Sudret

Publications and source records attributed to Bruno Sudret.

At least 19 recordsLinked to original sources

arXiv2026

Active learning strategy for excursion-set confidence regions of functional simulator outputs

Estimating excursion set confidence regions seeks to identify regions where a function may exceed some threshold with a given confidence level. This paper focuses on estimating such confidence regions in cases where the function has random inputs and a functional output that is returned all at once. We develop a surrogate-based approach for estimating the confidence region, combining principal component analysis and Gaussian process regression. An active learning strategy is also introduced, based on a max-min criterion that selects new samples which are likely to reduce the uncertainty in the confidence region. This strategy leverages efficient sampling of the Gaussian process through a Karhunen-Lo\`eve expansion. The proposed approach is applied to estimate the confidence regions of three case studies: a synthetic function, the surface pressure coefficient distribution of a hypersonic vehicle, and the glide-back trajectory of a reusable launcher first stage. The method demonstrates efficiency in accurately estimating the confidence region while reducing sources of modeling uncertainties. It is benchmarked against reference methods from the literature. Relevant metrics for assessing the confidence region estimation performance are discussed.

stat.ME↗

arXiv2026

Time-variant reliability using time-dependent surrogate models

Time-variant reliability analysis is a critical task for ensuring the safety of engineering dynamical systems subjected to stochastic excitations. However, assessing failure probability for realistic systems with Monte-Carlo simulation-based methods is often computationally intractable due to the high cost of the underlying models and the large number of simulations required. While surrogate models such as polynomial chaos expansions or Kriging are well-established for time-invariant reliability problems, their direct application to time-dependent systems remains challenging. This chapter introduces two advanced surrogate modeling frameworks designed specifically for dynamical systems: manifold-NARX (mNARX) and functional NARX (F-NARX). The mNARX approach constructs the surrogate on a reduced-order manifold of auxiliary state variables, enabling the efficient handling of high-dimensional inputs by embedding physical insight into a regression formulation. Conversely, the F-NARX framework exploits the functional nature of system trajectories, extracting principal component features from continuous time windows to mitigate issues associated with discrete lag selection and long-memory effects. We demonstrate the efficacy of these methods on two benchmark reliability problems: a stochastic quarter-car model and a hysteretic Bouc-Wen oscillator. The results highlight that, when combined with suitably biased experimental designs, both frameworks accurately capture the tail behavior of the system response, enabling precise and efficient estimation of first-passage probabilities.

stat.ME↗

arXiv2026

Probabilistic function-on-function nonlinear autoregressive model for emulation and reliability analysis of stochastic dynamical systems

Constructing accurate and computationally efficient surrogate models (or emulators) for predicting dynamical system responses is critical in many engineering domains, yet remains challenging due to the strongly nonlinear and high-dimensional mapping from external excitations and system parameters to system responses. This work introduces a novel Function-on-Function Nonlinear AutoRegressive model with eXogenous inputs (F2NARX), which reformulates the recently proposed $\mathcal{F}$-NARX method from a function-on-function regression perspective. The proposed framework substantially improves predictive efficiency while maintaining high accuracy. By combining principal component analysis with Gaussian process regression, F2NARX further enables probabilistic predictions of dynamical responses via the unscented transform in an autoregressive manner. Such probabilistic prediction capabilities further facilitate active learning for first-passage probability evaluation. The effectiveness of the method is demonstrated through case studies of varying complexity. Results show that F2NARX outperforms state-of-the-art NARX model by orders of magnitude in efficiency while achieving higher accuracy in general. Meanwhile, the active learning approach enables accurate estimation of first-passage failure probabilities for dynamical systems using only a small number of training time histories.

math.DS↗

arXiv2024

Reliability analysis for non-deterministic limit-states using stochastic emulators

Reliability analysis is a sub-field of uncertainty quantification that assesses the probability of a system performing as intended under various uncertainties. Traditionally, this analysis relies on deterministic models, where experiments are repeatable, i.e., they produce consistent outputs for a given set of inputs. However, real-world systems often exhibit stochastic behavior, leading to non-repeatable outcomes. These so-called stochastic simulators produce different outputs each time the model is run, even with fixed inputs. This paper formally introduces reliability analysis for stochastic models and addresses it by using suitable surrogate models to lower its typically high computational cost. Specifically, we focus on the recently introduced generalized lambda models and stochastic polynomial chaos expansions. These emulators are designed to learn the inherent randomness of the simulator's response and enable efficient uncertainty quantification at a much lower cost than traditional Monte Carlo simulation. We validate our methodology through three case studies. First, using an analytical function with a closed-form solution, we demonstrate that the emulators converge to the correct solution. Second, we present results obtained from the surrogates using a toy example of a simply supported beam. Finally, we apply the emulators to perform reliability analysis on a realistic wind turbine case study, where only a dataset of simulation results is available.

stat.CO↗

arXiv2024

Surrogate modeling with functional nonlinear autoregressive models (F-NARX)

We propose a novel functional approach to surrogate modeling of dynamical systems with exogenous inputs. This approach, named Functional Nonlinear AutoRegressive with eXogenous inputs (F-NARX), approximates the system response based on temporal features of the exogenous inputs and the system response. This marks a major step away from the discrete-time-centric approach of classical NARX models, which determines the relationship between selected time steps of the input/output time series. By modeling the system in a time-feature space, F-NARX takes advantage of the temporal smoothness of the process being modeled, providing more stable predictions and reducing the dependence of model performance on the discretization of the time axis. In this work, we introduce an F-NARX implementation based on principal component analysis and polynomial regression. To further improve prediction accuracy, we also introduce a modified hybrid least angle regression approach to identify a sparse model structure and minimize the expected forecast error, rather than the one-step-ahead prediction error. We investigate the behavior and capabilities of our F-NARX implementation on two case studies: an eight-story building under wind loading and a three-story steel frame under seismic loading. Our results demonstrate that F-NARX has several favorable properties that make it well-suited to surrogate modeling applications.

stat.ME↗

arXiv2024

A survey on multi-fidelity surrogates for simulators with functional outputs: unified framework and benchmark

Multi-fidelity surrogate models combining dimensionality reduction and an intermediate surrogate in the reduced space allow a cost-effective emulation of simulators with functional outputs. The surrogate is an input-output mapping learned from a limited number of simulator evaluations. This computational efficiency makes surrogates commonly used for many-query tasks. Diverse methods for building them have been proposed in the literature, but they have only been partially compared. This paper introduces a unified framework encompassing the different surrogate families, followed by a methodological comparison and the exposition of practical considerations. More than a dozen of existing multi-fidelity surrogates have been implemented under the unified framework and evaluated on a set of benchmark problems. Based on the results, guidelines and recommendations are proposed regarding multi-fidelity surrogates with functional outputs. Our study shows that most multi-fidelity surrogates outperform their tested single-fidelity counterparts under the considered settings. But no particular surrogate is performing better on every test case. Therefore, the selection of a surrogate should consider the specific properties of the emulated functions, in particular the correlation between the low- and high-fidelity simulators, the size of the training set, the local nonlinear variations in the residual fields, and the size of the training datasets.

cs.CE↗

arXiv2024

Maximum a Posteriori Estimation for Linear Structural Dynamics Models Using Bayesian Optimization with Rational Polynomial Chaos Expansions

Bayesian analysis enables combining prior knowledge with measurement data to learn model parameters. Commonly, one resorts to computing the maximum a posteriori (MAP) estimate, when only a point estimate of the parameters is of interest. We apply MAP estimation in the context of structural dynamic models, where the system response can be described by the frequency response function. To alleviate high computational demands from repeated expensive model calls, we utilize a rational polynomial chaos expansion (RPCE) surrogate model that expresses the system frequency response as a rational of two polynomials with complex coefficients. We propose an extension to an existing sparse Bayesian learning approach for RPCE based on Laplace's approximation for the posterior distribution of the denominator coefficients. Furthermore, we introduce a Bayesian optimization approach, which allows to adaptively enrich the experimental design throughout the optimization process of MAP estimation. Thereby, we utilize the expected improvement acquisition function as a means to identify sample points in the input space that are possibly associated with large objective function values. The acquisition function is estimated through Monte Carlo sampling based on the posterior distribution of the expansion coefficients identified in the sparse Bayesian learning process. By combining the sparsity-inducing learning procedure with the sequential experimental design, we effectively reduce the number of model evaluations in the MAP estimation problem. We demonstrate the applicability of the presented methods on the parameter updating problem of an algebraic two-degree-of-freedom system and the finite element model of a cross-laminated timber plate.

stat.ML↗

arXiv2024

Reliability analysis of arbitrary systems based on active learning and global sensitivity analysis

System reliability analysis aims at computing the probability of failure of an engineering system given a set of uncertain inputs and limit state functions. Active-learning solution schemes have been shown to be a viable tool but as of yet they are not as efficient as in the context of component reliability analysis. This is due to some peculiarities of system problems, such as the presence of multiple failure modes and their uneven contribution to failure, or the dependence on the system configuration (e.g., series or parallel). In this work, we propose a novel active learning strategy designed for solving general system reliability problems. This algorithm combines subset simulation and Kriging/PC-Kriging, and relies on an enrichment scheme tailored to specifically address the weaknesses of this class of methods. More specifically, it relies on three components: (i) a new learning function that does not require the specification of the system configuration, (ii) a density-based clustering technique that allows one to automatically detect the different failure modes, and (iii) sensitivity analysis to estimate the contribution of each limit state to system failure so as to select only the most relevant ones for enrichment. The proposed method is validated on two analytical examples and compared against results gathered in the literature. Finally, a complex engineering problem related to power transmission is solved, thereby showcasing the efficiency of the proposed method in a real-case scenario.

stat.ME↗

arXiv2024

Reliability analysis for data-driven noisy models using active learning

Reliability analysis aims at estimating the failure probability of an engineering system. It often requires multiple runs of a limit-state function, which usually relies on computationally intensive simulations. Traditionally, these simulations have been considered deterministic, i.e., running them multiple times for a given set of input parameters always produces the same output. However, this assumption does not always hold, as many studies in the literature report non-deterministic computational simulations (also known as noisy models). In such cases, running the simulations multiple times with the same input will result in different outputs. Similarly, data-driven models that rely on real-world data may also be affected by noise. This characteristic poses a challenge when performing reliability analysis, as many classical methods, such as FORM and SORM, are tailored to deterministic models. To bridge this gap, this paper provides a novel methodology to perform reliability analysis on models contaminated by noise. In such cases, noise introduces latent uncertainty into the reliability estimator, leading to an incorrect estimation of the real underlying reliability index, even when using Monte Carlo simulation. To overcome this challenge, we propose the use of denoising regression-based surrogate models within an active learning reliability analysis framework. Specifically, we combine Gaussian process regression with a noise-aware learning function to efficiently estimate the probability of failure of the underlying noise-free model. We showcase the effectiveness of this methodology on standard benchmark functions and a finite element model of a realistic structural frame.

stat.CO↗

arXiv2024

Uncertainty-aware multi-fidelity surrogate modeling with noisy data

Emulating high-accuracy computationally expensive models is crucial for tasks requiring numerous model evaluations, such as uncertainty quantification and optimization. When lower-fidelity models are available, they can be used to improve the predictions of high-fidelity models. Multi-fidelity surrogate models combine information from sources of varying fidelities to construct an efficient surrogate model. However, in real-world applications, uncertainty is present in both high- and low-fidelity models due to measurement or numerical noise, as well as lack of knowledge due to the limited experimental design budget. This paper introduces a comprehensive framework for multi-fidelity surrogate modeling that handles noise-contaminated data and is able to estimate the underlying noise-free high-fidelity model. Our methodology quantitatively incorporates the different types of uncertainty affecting the problem and emphasizes on delivering precise estimates of the uncertainty in its predictions both with respect to the underlying high-fidelity model and unseen noise-contaminated high-fidelity observations, presented through confidence and prediction intervals, respectively. Additionally, the proposed framework offers a natural approach to combining physical experiments and computational models by treating noisy experimental data as high-fidelity sources and white-box computational models as their low-fidelity counterparts. The effectiveness of our methodology is showcased through synthetic examples and a wind turbine application.

stat.ME↗

arXiv2023

Emulating the dynamics of complex systems using autoregressive models on manifolds (mNARX)

We propose a novel surrogate modelling approach to efficiently and accurately approximate the response of complex dynamical systems driven by time-varying exogenous excitations over extended time periods. Our approach, namely manifold nonlinear autoregressive modelling with exogenous input (mNARX), involves constructing a problem-specific exogenous input manifold that is optimal for constructing autoregressive surrogates. The manifold, which forms the core of mNARX, is constructed incrementally by incorporating the physics of the system, as well as prior expert- and domain- knowledge. Because mNARX decomposes the full problem into a series of smaller sub-problems, each with a lower complexity than the original, it scales well with the complexity of the problem, both in terms of training and evaluation costs of the final surrogate. Furthermore, mNARX synergizes well with traditional dimensionality reduction techniques, making it highly suitable for modelling dynamical systems with high-dimensional exogenous inputs, a class of problems that is typically challenging to solve. Since domain knowledge is particularly abundant in physical systems, such as those found in civil and mechanical engineering, mNARX is well suited for these applications. We demonstrate that mNARX outperforms traditional autoregressive surrogates in predicting the response of a classical coupled spring-mass system excited by a one-dimensional random excitation. Additionally, we show that mNARX is well suited for emulating very high-dimensional time- and state-dependent systems, even when affected by active controllers, by surrogating the dynamics of a realistic aero-servo-elastic onshore wind turbine simulator. In general, our results demonstrate that mNARX offers promising prospects for modelling complex dynamical systems, in terms of accuracy and efficiency.

stat.CO↗

arXiv2023

Global sensitivity analysis using derivative-based sparse Poincaré chaos expansions

Variance-based global sensitivity analysis, in particular Sobol' analysis, is widely used for determining the importance of input variables to a computational model. Sobol' indices can be computed cheaply based on spectral methods like polynomial chaos expansions (PCE). Another choice are the recently developed Poincaré chaos expansions (PoinCE), whose orthonormal tensor-product basis is generated from the eigenfunctions of one-dimensional Poincaré differential operators. In this paper, we show that the Poincaré basis is the unique orthonormal basis with the property that partial derivatives of the basis form again an orthogonal basis with respect to the same measure as the original basis. This special property makes PoinCE ideally suited for incorporating derivative information into the surrogate modelling process. Assuming that partial derivative evaluations of the computational model are available, we compute spectral expansions in terms of Poincaré basis functions or basis partial derivatives, respectively, by sparse regression. We show on two numerical examples that the derivative-based expansions provide accurate estimates for Sobol' indices, even outperforming PCE in terms of bias and variance. In addition, we derive an analytical expression based on the PoinCE coefficients for a second popular sensitivity index, the derivative-based sensitivity measure (DGSM), and explore its performance as upper bound to the corresponding total Sobol' indices.

stat.CO↗

arXiv2022

A spectral surrogate model for stochastic simulators computed from trajectory samples

Stochastic simulators are non-deterministic computer models which provide a different response each time they are run, even when the input parameters are held at fixed values. They arise when additional sources of uncertainty are affecting the computer model, which are not explicitly modeled as input parameters. The uncertainty analysis of stochastic simulators requires their repeated evaluation for different values of the input variables, as well as for different realizations of the underlying latent stochasticity. The computational cost of such analyses can be considerable, which motivates the construction of surrogate models that can approximate the original model and its stochastic response, but can be evaluated at much lower cost. We propose a surrogate model for stochastic simulators based on spectral expansions. Considering a certain class of stochastic simulators that can be repeatedly evaluated for the same underlying random event, we view the simulator as a random field indexed by the input parameter space. For a fixed realization of the latent stochasticity, the response of the simulator is a deterministic function, called trajectory. Based on samples from several such trajectories, we approximate the latter by sparse polynomial chaos expansion and compute analytically an extended Karhunen-Loève expansion (KLE) to reduce its dimensionality. The uncorrelated but dependent random variables of the KLE are modeled by advanced statistical techniques such as parametric inference, vine copula modeling, and kernel density estimation. The resulting surrogate model approximates the marginals and the covariance function, and allows to obtain new realizations at low computational cost. We observe that in our numerical examples, the first mode of the KLE is by far the most important, and investigate this phenomenon and its implications.

stat.CO↗

arXiv2022

Sequential active learning of low-dimensional model representations for reliability analysis

To date, the analysis of high-dimensional, computationally expensive engineering models remains a difficult challenge in risk and reliability engineering. We use a combination of dimensionality reduction and surrogate modelling termed partial least squares-driven polynomial chaos expansion (PLS-PCE) to render such problems feasible. Standalone surrogate models typically perform poorly for reliability analysis. Therefore, in a previous work, we have used PLS-PCEs to reconstruct the intermediate densities of a sequential importance sampling approach to reliability analysis. Here, we extend this approach with an active learning procedure that allows for improved error control at each importance sampling level. To this end, we formulate an estimate of the combined estimation error for both the subspace identified in the dimension reduction step and surrogate model constructed therein. With this, it is possible to adapt the design of experiments so as to optimally learn the subspace representation and the surrogate model constructed therein. The approach is gradient-free and thus can be directly applied to black box-type models. We demonstrate the performance of this approach with a series of low- (2 dimensions) to high- (869 dimensions) dimensional example problems featuring a number of well-known caveats for reliability methods besides high dimensions and expensive computational models: strongly nonlinear limit-state functions, multiple relevant failure regions and small probabilities of failure.

stat.CO↗

arXiv2021

Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applications

Sparse polynomial chaos expansions (PCE) are an efficient and widely used surrogate modeling method in uncertainty quantification for engineering problems with computationally expensive models. To make use of the available information in the most efficient way, several approaches for so-called basis-adaptive sparse PCE have been proposed to determine the set of polynomial regressors ("basis") for PCE adaptively. The goal of this paper is to help practitioners identify the most suitable methods for constructing a surrogate PCE for their model. We describe three state-of-the-art basis-adaptive approaches from the recent sparse PCE literature and conduct an extensive benchmark in terms of global approximation accuracy on a large set of computational models. Investigating the synergies between sparse regression solvers and basis adaptivity schemes, we find that the choice of the proper solver and basis-adaptive scheme is very important, as it can result in more than one order of magnitude difference in performance. No single method significantly outperforms the others, but dividing the analysis into classes (regarding input dimension and experimental design size), we are able to identify specific sparse solver and basis adaptivity combinations for each class that show comparatively good performance. To further improve on these findings, we introduce a novel solver and basis adaptivity selection scheme guided by cross-validation error. We demonstrate that this automatic selection procedure provides close-to-optimal results in terms of accuracy, and significantly more robust solutions, while being more general than the case-by-case recommendations obtained by the benchmark.

stat.CO↗

arXiv2021

Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark

Sparse polynomial chaos expansions (PCE) are a popular surrogate modelling method that takes advantage of the properties of PCE, the sparsity-of-effects principle, and powerful sparse regression solvers to approximate computer models with many input parameters, relying on only few model evaluations. Within the last decade, a large number of algorithms for the computation of sparse PCE have been published in the applied math and engineering literature. We present an extensive review of the existing methods and develop a framework for classifying the algorithms. Furthermore, we conduct a unique benchmark on a selection of methods to identify which approaches work best in practical applications. Comparing their accuracy on several benchmark models of varying dimensionality and complexity, we find that the choice of sparse regression solver and sampling scheme for the computation of a sparse PCE surrogate can make a significant difference, of up to several orders of magnitude in the resulting mean-squared error. Different methods seem to be superior in different regimes of model dimensionality and experimental design size.

math.NA↗

arXiv2019

Surrogate modeling of indoor down-link human exposure based on sparse polynomial chaos expansion

Human exposure induced by wireless communication systems increasingly draws the public attention. Here, an indoor down-link scenario is concerned and the exposure level is statistically analyzed. The electromagnetic field (EMF) emitted by a WiFi box is measured and electromagnetic dosimetry features are evaluated from the whole-body specific absorption rate as computed with a Finite-Difference Time-Domain (a.k.a. FDTD) code. Due to computational cost, a statistical analysis is performed based on a surrogate model, which is constructed by means of so-called sparse polynomial chaos expansion (PCE), where the inner cross validation (ICV) is used to select the optimal hyperparameters during the model construction and assess the model performance. However, the ICV error is optimized and the model assessment tends to be overly optimistic with small data sets. The method of cross-model validation is used and outer cross validation is carried out for the model assessment. The effects of the data preprocessing are investigated as well. Based on the surrogate model, the global sensitivity of the exposure to input parameters is analyzed from Sobol' indices.

stat.AP↗

arXiv2019

Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: application to urban drainage simulation

This paper presents an efficient surrogate modeling strategy for the uncertainty quantification and Bayesian calibration of a hydrological model. In particular, a process-based dynamical urban drainage simulator that predicts the discharge from a catchment area during a precipitation event is considered. The goal of the case study is to perform a global sensitivity analysis and to identify the unknown model parameters as well as the measurement and prediction errors. These objectives can only be achieved by cheapening the incurred computational costs, that is, lowering the number of necessary model runs. With this in mind, a regularity-exploiting metamodeling technique is proposed that enables fast uncertainty quantification. Principal component analysis is used for output dimensionality reduction and sparse polynomial chaos expansions are used for the emulation of the reduced outputs. Sobol' sensitivity indices are obtained directly from the expansion coefficients by a mere post-processing. Bayesian inference via Markov chain Monte Carlo posterior sampling is drastically accelerated.

stat.CO↗