arXiv subjects

Matthias G. R. Faes

Publications and source records attributed to Matthias G. R. Faes.

6 recordsLinked to original sources

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↗

arXiv2025

On the impact of geometric variance on the performance of formed parts: A probabilistic approach on the example of airbag pressure bins

Scatter in properties resulting from manufacturing is a great challenge in lightweight design, requiring consideration of not only the average mechanical performance but also the variance which is done e.g., by conservative safety factors. One contributor to this variance is the inherent geometric variability in the formed part. To isolate and quantify this effect, we present a probabilistic numerical study, aiming to assess the impact of geometric variance on the resulting part performance. By modelling geometric deviations stochastically, we aim to establish a correlation between the variance in geometry with the resulting variance in performance. The study is done on the example of an airbag pressure bin, where a better understanding of this correlation is crucial, as it allows for the design of a lighter part without changing the manufacturing process. Instead, we aim to implement more targeted and effective quality assurance, informed by the performance impact of geometric deviations.

cs.CE↗

arXiv2025

Bayesian model updating via streamlined Bayesian active learning cubature

This paper proposes a novel Bayesian active learning method for Bayesian model updating, which is termed as "Streamlined Bayesian Active Learning Cubature" (SBALC). The core idea is to approximate the log-likelihood function using Gaussian process (GP) regression in a streamlined Bayesian active learning way. Rather than generating many samples from the posterior GP, we only use its mean and variance function to form the model evidence estimator, stopping criterion, and learning function. Specifically, the estimation of model evidence is first treated as a Bayesian cubature problem, with a GP prior assigned over the log-likelihood function. Second, a plug-in estimator for model evidence is proposed based on the posterior mean function of the GP. Third, an upper bound on the expected absolute error between the posterior model evidence and its plug-in estimator is derived. Building on this result, a novel stopping criterion and learning function are proposed using only the posterior mean and standard deviation functions of the GP. Finally, we can obtain the model evidence based on the posterior mean function of the log-likelihood function in conjunction with Monte Carlo simulation, as well as the samples for the posterior distribution of model parameters as a by-product. Four numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method compared to several existing approaches. The results show that the method can significantly reduce the number of model evaluations and the computational time without compromising accuracy.

stat.CO↗

arXiv2025

Efficient forward and inverse uncertainty quantification for dynamical systems based on dimension reduction and Kriging surrogate modeling in functional space

Surrogate models are extensively employed for forward and inverse uncertainty quantification in complex, computation-intensive engineering problems. Nonetheless, constructing high-accuracy surrogate models for complex dynamical systems with limited training samples continues to be a challenge, as capturing the variability in high-dimensional dynamical system responses with a small training set is inherently difficult. This study introduces an efficient Kriging modeling framework based on functional dimension reduction (KFDR) for conducting forward and inverse uncertainty quantification in dynamical systems. By treating the responses of dynamical systems as functions of time, the proposed KFDR method first projects these responses onto a functional space spanned by a set of predefined basis functions, which can deal with noisy data by adding a roughness regularization term. A few key latent functions are then identified by solving the functional eigenequation, mapping the time-variant responses into a low-dimensional latent functional space. Subsequently, Kriging surrogate models with noise terms are constructed in the latent space. With an inverse mapping established from the latent space to the original output space, the proposed approach enables accurate and efficient predictions for dynamical systems. Finally, the surrogate model derived from KFDR is directly utilized for efficient forward and inverse uncertainty quantification of the dynamical system. Through three numerical examples, the proposed method demonstrates its ability to construct highly accurate surrogate models and perform uncertainty quantification for dynamical systems accurately and efficiently.

math.DS↗

arXiv2024

Response probability distribution estimation of expensive computer simulators: A Bayesian active learning perspective using Gaussian process regression

Estimation of the response probability distributions of computer simulators in the presence of randomness is a crucial task in many fields. However, achieving this task with guaranteed accuracy remains an open computational challenge, especially for expensive-to-evaluate computer simulators. In this work, a Bayesian active learning perspective is presented to address the challenge, which is based on the use of the Gaussian process (GP) regression. First, estimation of the response probability distributions is conceptually interpreted as a Bayesian inference problem, as opposed to frequentist inference. This interpretation provides several important benefits: (1) it quantifies and propagates discretization error probabilistically; (2) it incorporates prior knowledge of the computer simulator, and (3) it enables the effective reduction of numerical uncertainty in the solution to a prescribed level. The conceptual Bayesian idea is then realized by using the GP regression, where we derive the posterior statistics of the response probability distributions in semi-analytical form and also provide a numerical solution scheme. Based on the practical Bayesian approach, a Bayesian active learning (BAL) method is further proposed for estimating the response probability distributions. In this context, the key contribution lies in the development of two crucial components for active learning, i.e., stopping criterion and learning function, by taking advantage of posterior statistics. It is empirically demonstrated by five numerical examples that the proposed BAL method can efficiently estimate the response probability distributions with desired accuracy.

stat.CO↗

arXiv2022

First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems by a fractional moments-based mixture distribution approach

First-passage probability estimation of high-dimensional nonlinear stochastic systems is a significant task to be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a novel approach, termed 'fractional moments-based mixture distribution', to address such challenge. This approach is implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional moments and mixture distribution. In our context, the fractional moment itself is by definition a high-dimensional integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS). In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments. From the knowledge of low-order fractional moments, the EVD of interest is then expected to be reconstructed. Based on introducing an extended inverse Gaussian distribution and a log extended skew-normal distribution, one flexible mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the first-passage probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance of the proposed method is verified by three examples consisting of two test examples and one engineering problem.

stat.ME↗