# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function
## 2 Surrogate Model for Function with Mono-Dimensional Domain Random Field Output
### Gaussian Process  
The definition of a surrogate model of \(\mathcal{Q}_{X}(\cdot)\) that maps the input uncertain variable vector \(\mathbf{U}\) to the stochastic process \(\mathbf{X}(t,\mathbf{U})\) with KL expansion, it is necessary to build the relation: \(\xi_{k}:\mathbf{u}\to\xi_{k}(\mathbf{u})\). In the previous model reduction, as the mapping is known only for the snapshots (i.e., limited number of samples \(\mathcal{U}_{M}\)) it is necessary to define a regression model to model \(\xi_{k}\) for all the values that can be taken by \(\mathbf{u}\). In the following, it is proposed to use Gaussian process as it is particularly adequate for uncertainty quantification [25]. Gaussian process (GP) [6; 7; 8] may be used as a stochastic process surrogate model in order to approximate any unknown function \(\xi_{k}(\cdot)\). The general idea is to consider that \(\xi_{k}(\cdot)\) is a sample path of a Gaussian process. In this section, to simplify the notations in the description of Gaussian process, the subscript \(k\) is removed. A GP encodes a distribution over a set of functions. A GP may be seen as a collection of an infinite number of random variables and any finite subset of this collection has a joint Gaussian distribution. A GP is fully determined by its mean and covariance functions. To build a GP \(\xi_{\ell}(\cdot)\), it is necessary to solve a supervised learning problem. Starting from an input training set of size \(M\), \(\mathcal{U}_{M}=\{\mathbf{u}_{1},\dots,\mathbf{u}_{M}\}\), the corresponding unknown function responses are obtained with Equation (5) and noted \(\mathcal{Y}_{M}=\{y_{1}=\xi(\mathbf{u}_{1}),\dots,\ y_{M}=\xi(\mathbf{u}_{M})\}\). In a regression context, a GP prior is assumed on the mean function \(m(\cdot)\) and on the covariance function \(k^{\Theta}(\mathbf{u},\mathbf{u}^{\prime})\). This covariance function depends on some hyper-parameters \(\Theta\). There exist different typical covariance functions (also called kernels) such as the p-exponential:  
\[k^{\Theta}(\mathbf{u},\mathbf{u}^{\prime})=\Theta_{\sigma}\exp\left(-\sum_{i=0 }^{d}\Theta_{\theta_{i}}|\mathbf{u}^{(i)}-\mathbf{u}^{\prime(i)}|^{\Theta_{Pi }}\right). \tag{15}\]  
The proposed strategy is adapted to any kernels. Regarding the mean function, as the tendency of the exact function is unknown, a constant function \(m\) is often assumed as GP prior, resulting in ordinary Kriging. Depending on available a priori knowledge, other types of mean function may be assumed such as quadratic or general basis functions. If a constant mean function is assumed, the GP is defined such that \(\xi(\cdot)\sim\mathcal{N}\big{(}m,k^{\Theta}(\cdot,\cdot)\big{)}\). Considering the Design Of Experiment (DoE) \(\Xi_{M}\), the GP has a multivariate Gaussian distribution \(\Xi_{M}|\mathcal{U}_{M}\sim\mathcal{N}\big{(}m,\mathbf{K}^{\Theta}_{MM}\big{)}\) where \(\mathbf{K}^{\Theta}_{MM}\) is the parameterized covariance function \(k^{\Theta}(\cdot)\) evaluated on \(\mathcal{U}_{M}\). The dependence on \(\Theta\) is dropped to simplify the notations. The prior assumption on the covariance function has an important influence on the type of modeled function by the GP. In the presence of experimental or numerical noisy data, the relationship between the latent function values \(\xi(\mathcal{U}_{M})\) and the observed responses \(\mathcal{Y}_{M}\) is given by: \(p(\mathbf{y}|\Xi_{M})=\mathcal{N}\big{(}\mathbf{y}|\Xi_{M},\sigma^{2}\mathbf{ I}\big{)}\) with \(\sigma^{2}\) an assumed Gaussian noise variance. Then, from this input and output training sets and the prior on the GP, it is possible to train it using the marginal likelihood. It is obtained by integrating out the latent function giving \(p(\mathbf{y}|\mathcal{U}_{M},\Theta)=\mathcal{N}\big{(}\mathbf{y}|m,\mathbf{ K}_{MM}+\sigma^{2}\mathbf{I}\big{)}\). To simplify the notations, we define \(\mathbf{\hat{K}}_{MM}=\mathbf{K}_{MM}+\sigma^{2}\mathbf{I}\). The GP training requires to maximize the log marginal likelihood with respect to the hyperparameters \(\Theta\), \(m\) and \(\sigma\).