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-Loève 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.

Functional covariates arise in many scientific and engineering applications when model inputs take the form of time-dependent or spatially distributed profiles, such as varying boundary conditions or changing material behaviours. In addition, new practices in digital simulation require predictions accompanied by confidence intervals. Models based on Gaussian processes (GPs) provide principled uncertainty quantification. However, GPs capable of jointly handling functional covariates and multiple correlated functional tasks remain largely under-explored. In this work, we extend the framework of GPs with functional covariates to multitask problems by introducing a fully separable kernel structure that captures dependencies across tasks and functional inputs. By taking advantage of the Kronecker structure of the covariance matrix, the model is made scalable. The proposed model is validated on a synthetic benchmark and applied to a realistic structure, a riveted assembly with functional descriptions of the material behaviour and response forces. The proposed functional multitask GP significantly improves over single task GPs. For the riveted assembly, it requires less than 100 samples to produce an accurate mean and confidence interval prediction. Despite its larger number of parameters, the multitask GP is computationally easier to learn than its single task pendant.

In this work, we propose a novel method to tackle the problem of multiobjective optimization under parameteric uncertainties, by considering the Conditional Pareto Sets and Conditional Pareto Fronts. Based on those quantities we can define the probability of coverage of the Conditional Pareto Set which can be interpreted as the probability for a design to be optimal in the Pareto sense. Due to the computational cost of such an approach, we introduce an Active Learning method based on Gaussian Process Regression in order to improve the estimation of this probability, which relies on a reformulation of the EHVI. We illustrate those methods on a few toy problems of moderate dimension, and on the problem of designing a cabin to highlight the differences in solutions brought by different formulations of the problem.

In this poster, we introduce a rank-based surrogate-assisted variant of CMA-ES. Unlike previous methods that employ rank information as constraints to train an SVM classifier, our approach employs a linear-quadratic regression on the ranks. We investigate the method's invariance empirically. While this first algorithm outperforms CMA-ES with a few exceptions, it falls short to entirely meet the lq-CMA-ES performance levels. To address this, we propose an enhanced variant that handles together two alternative surrogates, one based on the ranks and one based on the original function values. Although this variant sacrifices strict invariance, it gains in robustness and achieves performance comparable to, or even exceeding, lq-CMA-ES on transformed problems. This last algorithm shows how simply incorporating new transformations of rank values could improve any surrogate-based CMA-ES variant.

Approximation of functions satisfying partial differential equations (PDEs) is paramount for simulation of physical fluid flows and other problems in physics. Recently, physics-informed machine learning approaches have proven useful as a data-driven complement to numerical models for partial differential equations, bringing faster responses and allowing us to capitalize on past observations. However, their efficiency and convergence depend on the availability of vast training datasets. For sparse observations, Gaussian process regression or Kriging has emerged as a powerful interpolation model, offering principled estimates and uncertainty quantification. Several attempts have been made to condition Gaussian processes on linear PDEs via artificial or collocation observations and kernel design.These methods suffer from scalability issues in higher dimensions and limited generalizability. The aim of this study is to explore the extension of the Kriging predictor in the presence of linear PDE information at a finite number of collocation points. Two approaches are proposed: 1) A collocated co-Kriging with primary observations of the physical field and auxiliary differential observations; 2) A constrained Kriging optimization problem strongly satisfying linear PDE constraints at the points of prediction through a Lagrangian formulation. Numerical experiments are given for ordinary differential equations, 2D harmonic PDEs and an application to perfect flows around a cylinder. This work highlights a trade-off between the computational efficiency of the Lagrange multipliers approach and the strict interpolation of observations.

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.

This paper introduces an evolutionary algorithm for objective functions defined over clouds of points of varying sizes. Such design variables are modeled as uniform discrete measures with finite support and the crossover and mutation operators of the algorithm are defined using the Wasserstein barycenter. We prove that the Wasserstein-based crossover has a contracting property in the sense that the support of the generated measure is included in the closed convex hull of the union of the two parents’ supports. We introduce boundary mutations to counteract this contraction. Variants of evolutionary operators based on Wasserstein barycenters are studied. We compare the resulting algorithm to a more classical, sequence-based, evolutionary algorithm on a family of test functions that include a wind-farm layout problem. The results show that Wasserstein-based evolutionary operators better capture the underlying geometrical structures of the considered test functions and outperform a reference evolutionary algorithm in the vast majority of the cases. The tests indicate that the mutation operators play a major part in the performances of the algorithms.

Bayesian optimization algorithms form an important class of methods to minimize functions that are costly to evaluate, which is a very common situation. These algorithms iteratively infer Gaussian processes from past observations of the function and decide where new observations should be made through the maximization of an acquisition criterion. Often, the objective function is defined on a compact set such as in a hyper-rectangle of the $d$-dimensional real space, and the bounds are chosen wide enough so that the optimum is inside the search domain. In this situation, this work provides a way to integrate in the acquisition criterion the \textit{a priori} information that these functions, once modeled as GP trajectories, should be evaluated at their minima, and not at any point as usual acquisition criteria do. We propose an adaptation of the widely used Expected Improvement acquisition criterion that accounts only for GP trajectories where the first order partial derivatives are zero and the Hessian matrix is positive definite. The new acquisition criterion keeps an analytical, computationally efficient, expression. This new acquisition criterion is found to improve Bayesian optimization on a test bed of functions made of Gaussian process trajectories in low dimension problems. The addition of first and second order derivative information is particularly useful for multimodal functions.

Quantization summarizes continuous distributions by calculating a discrete approximation. Among the widely adopted methods for data quantization is Lloyd's algorithm, which partitions the space into Vorono\"i cells, that can be seen as clusters, and constructs a discrete distribution based on their centroids and probabilistic masses. Lloyd's algorithm estimates the optimal centroids in a minimal expected distance sense, but this approach poses significant challenges in scenarios where data evaluation is costly, and relates to rare events. Then, the single cluster associated to no event takes the majority of the probability mass. In this context, a metamodel is required and adapted sampling methods are necessary to increase the precision of the computations on the rare clusters.

We consider the problem of learning time-consuming functions defined over unordered sets of vectors.Such functions arise frequently, in particular in the context of networks of devices whose number is not fixed and that interact with each other.A working example is the modeling of a wind farm.Unordered sets of vectors are a mix of integer and continuous input variables suitable for functions that are permutation-invariant.The time-consuming aspect of the functions is, classically, treated by approximating them with a Gaussian process.This study addresses the problem of defining valid and efficient covariance kernels over clouds of points in the context of Gaussian process surrogate modeling.We review methods for defining such kernels.These kernels are compared on a set of analytical functions inspired from different engineering problems, such as the design of experiments and the modeling of wind farms production.The extrapolation properties of the kernels are tested on geometrically transformed clouds.We show that modeling 2D clouds of points as supports of discrete uniform distributions should be preferred to a Gaussian representation of the clouds.A detailed investigation of the good performance of MMD-based kernels illustrates how they adapt their hyperparameters to the geometrical properties of the studied functions.

We consider Reliability-based Robust Design Optimization (RRDO) where it is sought to optimize the mean of an objective function while satisfying constraints in probability. The high computational cost of the simulations underlying the objective and constraints strongly limits the number of evaluations and makes this type of problems particularly challenging. The numerical cost issue and the parametric uncertainties have been addressed with Bayesian optimization algorithms which leverage Gaussian processes of the objective and constraint functions. Current Bayesian optimization algorithms call the objective and constraint functions simultaneously at each iteration. This is often not necessary and overlooks calculation savings opportunity. This article proposes a new efficient RRDO Bayesian optimization algorithm that optimally selects for evaluation, not only the usual design variables, but also one or several constraints along with the uncertain parameters. The algorithm relies on a multi-output Gaussian model of the constraints. The coupling of constraints and their separated selection are gradually implemented in three algorithm variants which are compared to a reference Bayesian approach. The results are promising in terms of convergence speed, accuracy and stability as observed on a two, a four and a 27-dimensional problem.

We present a surrogate-assisted multiobjective optimization algorithm. The aggregation of the objectives relies on the Uncrowded Hypervolume Improvement (UHVI) which is partly replaced by a linear-quadratic surrogate that is integrated into the CMA-ES algorithm. Surrogating the UHVI poses two challenges. First, the UHVI is a dynamic function, changing with the empirical Pareto set. Second, it is a composite function, defined differently for dominated and nondominated points. The presented algorithm is thought to be used with expensive functions of moderate dimension (up to about 50) with a quadratic surrogate which is updated based on its ranking ability. We report numerical experiments which include tests on the COCO benchmark. The algorithm shows in particular linear convergence on the double sphere function with a convergence rate that is 6--20 times faster than without surrogate assistance.

The investigation of mixture models is a key to understand and visualize the distribution of multivariate data. Most mixture models approaches are based on likelihoods, and are not adapted to distribution with finite support or without a well-defined density function. This study proposes the Augmented Quantization method, which is a reformulation of the classical quantization problem but which uses the p-Wasserstein distance. This metric can be computed in very general distribution spaces, in particular with varying supports. The clustering interpretation of quantization is revisited in a more general framework. The performance of Augmented Quantization is first demonstrated through analytical toy problems. Subsequently, it is applied to a practical case study involving river flooding, wherein mixtures of Dirac and Uniform distributions are built in the input space, enabling the identification of the most influential variables.

Identifying model parameters is nowadays intrinsically linked with quantifying the associated uncertainties. While classical methods allow to handle some types of uncertainties such as experimental noise, they are not designed to take into account the variability between the different test specimens, significant in particular for composites materials. The estimation of the impact of this intrinsic variability on the material properties can be achieved using population approaches where this variability is modeled by a probability distribution (e.g., a multivariate Gaussian distribution). The objective is to calibrate this distribution (or equivalently its parameters for a parametric distribution). Among the estimation methods can be found mixed-effects models where the parameters that characterize each replication are decomposed between the population averaged behavior (called fixed-effects) and the impact of material variability (called random-effects). Yet, when the number of model parameters or the computational time of a single run of the simulations increases (for multiaxial models for instance), the simultaneous, global identification of all the material parameters is difficult because of the number of unknown quantities to estimate and because of the required model evaluations. Furthermore, the parameters do not have the same influence on the material constitutive model depending for instance on the nature of the load (e.g., tension, compression). The method proposed in this paper enables to calibrate the model on multiple experiments. It decomposes the overall calibration problem into a sequence of calibrations, each subproblem allowing to calibrate the joint distribution of a subset of the model parameters. The calibration process is eased as the number as the number of unknown parameters is reduced compared to the full problem. The proposed calibration process is applied to an orthotropic elastic model with non linear longitudinal behavior, for a unidirectional composite ply made of carbon fibers and epoxy resin. The ability of the method to sequentially estimate the model parameters distribution is investigated. Its capability to ensure consistency throughout the calibration process is also discussed. Results show that the methodology allows to handle the calibration of complex material constitutive models in the mixed-effects framework.

Visualization is an essential operation when assessing the risk of rare events such as coastal or river floodings. The goal is to display a few prototype events that best represent the probability law of the observed phenomenon, a task known as quantization. It becomes a challenge when data is expensive to generate and critical events are scarce, like extreme natural hazard. In the case of floodings, each event relies on an expensive-to-evaluate hydraulic simulator which takes as inputs offshore meteo-oceanic conditions and dyke breach parameters to compute the water level map. In this article, Lloyd's algorithm, which classically serves to quantize data, is adapted to the context of rare and costly-to-observe events. Low probability is treated through importance sampling, while Functional Principal Component Analysis combined with a Gaussian process deal with the costly hydraulic simulations. The calculated prototype maps represent the probability distribution of the flooding events in a minimal expected distance sense, and each is associated to a probability mass. The method is first validated using a 2D analytical model and then applied to a real coastal flooding scenario. The two sources of error, the metamodel and the importance sampling, are evaluated to quantify the precision of the method.