# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 1 Introduction  
Uncertainty quantification is often used to provide confidence measures over results obtained in numerical studies involving simulation modeling tools. For complex simulation codes, uncertainty quantification often requires to use uncertainty propagation techniques on a black-box function to estimate uncertainty measures on the function outputs considering an input joint distribution. Therefore, Monte Carlo simulation may be used to estimate uncertainty measures (e.g., expected value, variance, quantile) by repeated evaluations of the black-box functions. However, when one evaluation of the black-box model is computationally intensive, such an uncertainty propagation technique is not affordable in a reasonable time frame. Alternative approaches relying on surrogate models (e.g., Polynomial Chaos Expansion [1; 2; 3; 4; 5], Gaussian Process [6; 7; 8]) may be used instead of the exact black-box function to reproduce the mappings between the input and output variables. In order to control the confidence in the estimation of the uncertainty measures of interest using the surrogate model instead of the exact model, active learning strategies have been proposed [9; 10; 11]. The general idea of such approaches is to adaptively update the dataset used to build the surrogate model according to an infill criterion. The infill criterion differs depending on the uncertainty measure of the black-box function output. For instance, active learning strategies have been proposed to estimate a probability of failure [11; 12; 13; 14] (probability of exceeding a threshold for the output), to determine a minimum of the function in the presence of uncertainty [15; 16; 17], or to assess an excursion set [18]. In the present paper, the considered output of the black-box function is a high-dimensional vector defined over a mono-dimensional domain (uni-variate random field that can be characterized by a stochastic process). In the presence of uncertainty, the output of the black-box model is a stochastic process. Such an output may result from computational fluid dynamics, finite element analysis or optimal control problem in which the output isdiscretized along a mono-dimensional time or space domain (e.g., altitude as a function of time, pressure distribution along a wing chord). In the presence of uncertainty, being able to determine a quantile of such an high dimensional output is essential to define for instance flight corridors for certification purposes. However, due to the computational cost of the black-box model and the high dimension of its output, the determination of a quantile of the considered output random field is challenging. In this paper, an active learning strategy is proposed to determine the quantile of the output stochastic process by combining model order reduction [19; 20; 21], Gaussian processes [6; 7; 8] and adaptive sampling strategy to control the confidence in the estimation of the quantile via an enrichment of the surrogate model. In a previous work [22], a surrogate model for the estimation of quantile of launch vehicle flight corridors has been proposed but without adaptive refinement of the surrogate model to control the error in the quantile estimation. The main contribution of this paper is the development of an active learning strategy allowing to efficiently reduce the uncertainty of the estimation of the time-dependent extreme quantile at an affordable computational cost for complex simulation problems.  
The rest of the paper is organized as follows. In Section 2, basics on model order reduction based on Karhunen-Loeve and Gaussian processes are recalled in the context of uncertainty propagation. In Section 3, the proposed active learning strategy is proposed with the dedicated surrogate model adaptive enrichment for quantile estimation of the output stochastic process. Eventually, in Section 4, the proposed approach is applied on an analytical test case and a launch vehicle trajectory problem representative of the challenges raised by the determination of flight corridors for launchers.