# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 2 Surrogate Model for Function with Mono-Dimensional Domain Random Field Output  
Let us consider a black-box model \(\mathcal{Q}_{\mathbf{X}}(\cdot)\):  
\[\mathbf{U}\sim\phi_{\mathbf{U}}\stackrel{{\mathcal{Q}_{\mathbf{X }}}}{{\longrightarrow}}\mathbf{X}(t,\mathbf{U}) \tag{1}\]  
which takes as input \(\mathbf{U}\) an aleatory vector of dimension \(d\) defined over a probability space \((\Omega,\mathcal{A},\mathbb{P})\) with \(\mathbf{Y}\) the universal set, \(\mathcal{A}\) a \(\sigma\)-algebra and \(\mathbb{P}\) the probability measure. \(\mathbf{U}\) is characterized by a joint probability density function (PDF) \(\phi_{\mathbf{U}}(\cdot)\) over its definition domain \(\Omega\). The output of \(\mathcal{Q}_{\mathbf{X}}(\cdot)\) is a realization of a univariate stochastic process \(\mathbf{X}:\mathcal{T}\times\Omega\rightarrow\mathbb{R}\) with \(\mathcal{T}\) a domain (could be a temporal or spatial domain) and \(\mathbf{X}(t,\mathbf{U})\in\mathbb{R}\). It is assumed that the domain \(\mathcal{T}\) is mono-dimensional. Without loss of generality, in the rest of the paper, a temporal domain is considered. Some notations may be introduced for the stochastic process \(\mathbf{X}(\cdot,\cdot)\). The aleatory variable \(\mathbf{X}_{t}:\Omega\rightarrow\mathbb{R}\) corresponds to the univariate stochastic process fixed at time \(t\in\mathcal{T}\) leading to: \(\mathbf{X}(t,\mathbf{u})=\mathbf{X}_{t}(\mathbf{u})\). A particular sample of the stochastic process for a given realization of \(\mathbf{U}\) is noted: \(\mathbf{X}(\mathbf{u}):\mathcal{T}\rightarrow\mathbb{R}\) and it corresponds to \(\mathbf{X}(\mathbf{u})(t)=\mathbf{X}(t,\mathbf{u})\). For two different times \(t_{0},t_{1}\in\mathcal{T}^{2}\), the autocovariance of the stochastic process is defined by \(C_{\mathbf{X}\mathbf{X}}(t_{0},t_{1})=\text{cov}[\mathbf{X}_{t0},\mathbf{X}_ {t1}]=\mathbb{E}[(\mathbf{X}_{t0}-\mu_{t0})(\mathbf{X}_{t1}-\mu_{t1})]\) with \(\mu_{tj}\) the mean associated to the stochastic process \(\mu_{t}=\mathbb{E}[\mathbf{X}_{t}]\). In practice, for engineering simulation problems, the evaluation of \(\mathcal{Q}_{\mathbf{X}}(\cdot)\) may be computationally intensive as it might require running a complex solver analysis (such as computational fluid dynamics, finite element analysis techniques or solving optimal control problem). Therefore, due to the computational cost, uncertainty propagation can be challenging and it might be difficult to estimate statistical moments or quantile of the resulting output stochastic process. The \(\eta\)-quantile over the stochastic process \(X(\cdot,\cdot)\) is defined by:  
\[q_{\eta}(t)=\inf_{v\in\mathbb{R}}\{\mathbb{P}[\mathbf{X}(t,\mathbf{U})\leq v ]\geq\eta\} \tag{2}\]  
For instance, for launch vehicle design, the determination of flight envelopes on the state variables (e.g., altitude, velocity, flight path angle) in the presence of uncertainty (e.g., modeling uncertainty on the rocket engine efficiency, on the structural masses) is of prime importance to define possible flight corridors. This flight envelope may be characterized by upper and lower quantile values over the stochastic process defining the state variable in the presence of uncertainty. In order to efficiently estimate quantiles on the output of \(\mathcal{Q}_{\mathbf{X}}(\cdot)\), a surrogate-based model approach is proposed in this paper. It combines three elements: model order reduction, Gaussian process and active learning strategy. As the output of the black-box function \(\mathcal{Q}_{\mathbf{X}}(\cdot)\) is a uni-variate field, a model order reduction technique is used to reduce the dimension of the output. Then, a mapping between the input variables and quantities of interest in the reduced dimension space is carried out using Gaussian process. It allows to define a surrogate model \(\hat{\mathcal{Q}}_{\mathbf{X}}(\cdot)\) of the exact application \(\mathcal{Q}_{\mathbf{X}}(\cdot)\).