# Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Surrogate Model for Function with Mono-Dimensional Domain Random Field Output

#### 2.1. Model Order Reduction

#### 2.2. Gaussian Process

#### 2.3. Quantile Estimation

- Simulate ${N}_{q}$ samples ${\mathbf{u}}_{1},\dots ,{\mathbf{u}}_{{N}_{q}}$ according to ${\varphi}_{\mathbf{U}}(\xb7)$
- Compute the empirical distribution function:$${\widehat{\mathsf{\Phi}}}_{\widehat{\mathbf{X}}(t,\mathbf{U})}\left(\mathbf{u}\right)=\frac{1}{{N}_{q}}\sum _{i=1}^{{N}_{q}}{\mathbb{I}}_{\widehat{\mathbf{X}}(t,{\mathbf{u}}_{i})\le \widehat{\mathbf{X}}(t,\mathbf{u})}$$
- Estimate the quantile ${\widehat{q}}_{\eta}\left(t\right)$ by$${\widehat{q}}_{\eta}\left(t\right)={\widehat{\mathsf{\Phi}}}_{\widehat{\mathbf{X}}(t,\mathbf{U})}^{-1}\left(\eta \right)$$

## 3. Active Learning for Surrogate-Based Stochastic Process Quantile Estimation

#### 3.1. Infill Criterion

Algorithm 1: Confidence interval area calculation for quantile estimation based on GP trajectories for a candidate sample $\tilde{\mathbf{u}}$ |

Inputs: ${\mathcal{U}}_{M},{\mathcal{Y}}_{M}$, $\tilde{\mathbf{u}}$, $\widehat{\mathbf{X}}(t,\mathbf{U})$ Initialization: $\eta $, $\beta $, ${N}_{q}$, ${N}_{GP}$ (1) Define new input and output “virtual” extended datasets ${\tilde{\mathcal{U}}}_{M+1}=\{{\mathbf{u}}_{1},\dots ,{\mathbf{u}}_{M},\tilde{\mathbf{u}}\}$ and ${\tilde{\mathcal{Y}}}_{{k}_{M+1}}=\left\{{y}_{1}={\xi}_{k}\left({\mathbf{u}}_{1}\right),\dots ,{y}_{M}={\xi}_{k}\left({\mathbf{u}}_{M}\right),{y}_{M+1}={\underline{\widehat{\xi}}}_{k}\left(\tilde{\mathbf{u}}\right)\right\}$ (2) Train the new “virtual” GPs ${\widehat{\xi}}_{k}^{*}(\xb7)$ based on ${\tilde{\mathcal{U}}}_{M+1}$ and ${\tilde{\mathcal{Y}}}_{{k}_{M+1}}$ (3) Generate ${N}_{GP}$ Gaussian process trajectories for the ${N}_{KL}$ GPs in KL $\left\{{\widehat{\xi}}_{k}^{*\left(1\right)}(\xb7),\dots ,{\widehat{\xi}}_{k}^{*\left(s\right)}(\xb7),\dots {\widehat{\xi}}_{k}^{*\left({N}_{GP}\right)}(\xb7)\right\}$ (4) for s = 1: ${N}_{GP}$ do$\phantom{(}$$\phantom{(}$end$\phantom{(}$(5) Define the set ${\mathbb{Q}}_{{N}_{GP}}=\left\{{\widehat{q}}_{\eta}^{\left(1\right)}\left(t\right),\dots ,{\widehat{q}}_{\eta}^{\left(s\right)}\left(t\right),\right.$$\left.\dots ,{\widehat{q}}_{\eta}^{\left({N}_{GP}\right)}\left(t\right)\right\}$ of ${N}_{GP}$ quantiles (6) Determine ${\overline{\widehat{q}}}_{\eta}\left(t\right)$ and ${\underline{\widehat{q}}}_{\eta}\left(t\right)$ as quantiles with $\beta $ and $1-\beta $ levels over the set ${\mathbb{Q}}_{{N}_{GP}}$. (7) Compute $A\left(\tilde{\mathbf{u}}\right)=\int \left({\overline{\widehat{q}}}_{\eta}\left(t\right)-{\underline{\widehat{q}}}_{\eta}\left(t\right)\right)\mathrm{d}t$ over the time mesh ${\mathcal{T}}_{{n}_{v}}$ return $A\left(\tilde{\mathbf{u}}\right)$ |

#### 3.2. Auxiliary Optimization Problem and Active Learning Process

## 4. Applications

#### 4.1. Analytical Test Case

#### 4.1.1. Test Case Description

#### 4.1.2. Test Case Results

#### 4.2. Launch Vehicle Design Problem

#### 4.2.1. Test Case Description

#### Propulsion

#### Structures

#### Aerodynamics

#### Trajectory

#### 4.2.2. Test Case Settings

#### 4.2.3. Results Analysis and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) output stochastic process realizations and 99% quantile prediction, (

**b**) the GPs involved in the surrogate model of the output stochastic process present uncertainty in the prediction and different GP realizations may be sampled (red curves), (

**c**) for each GP realization, a quantile of the output stochastic process may be estimated (black dot lines), (

**d**) based on the set of quantile estimations an confidence interval on the exact quantile may be estimated.

**Figure 3.**Proposed active learning strategy for quantile estimation refinement for stochastic process defined over a mono-dimensional mesh domain.

**Figure 6.**Karhunen–Loève validation with respect to the number of modes: 1 (

**a**), 2 (

**b**), 4 (

**c**), 6 (

**d**).

**Figure 9.**Estimation of the quantile and its confidence area before refinement (

**a**) and after 10 active-learning iterations (

**b**).

**Figure 10.**Boxplots for 10 repetitions of the estimation of the quantile confidence area (

**a**) and root mean square error (

**b**) along the 10 active-learning iterations.

**Figure 11.**Confidence area based on the surrogate model estimation and based on Monte Carlo (MC) with 14 evaluations of ${\mathcal{Q}}_{\mathbf{X}}(\xb7)$.

**Figure 12.**Uncertainty propagation for launch vehicle design test case using multidisciplinary simulation.

**Figure 13.**Visualization of the launch vehicle ascent trajectories in 3D due to the uncertain input vector.

**Figure 15.**Karhunen–Loève validation with respect to the number of modes: 2 (

**a**), 5 (

**b**), 10 (

**c**), 20 (

**d**) and 40 (

**e**).

**Figure 16.**Initial and final output stochastic process samples for the altitude (

**a**,

**c**) and the centered altitude (

**b**,

**d**).

**Figure 18.**Quantile estimation and confidence area with initial dataset (

**a**,

**c**) and after 10 active-learning-based enriched samples (

**b**,

**d**).

**Figure 19.**Estimation of the quantile confidence area (

**a**) and root mean square error (

**b**) along the 10 active-learning iterations for 10 repetitions.

Number of KL Modes | 1 | 2 | 4 | 6 |
---|---|---|---|---|

Q2 | −489.28 | 0.9998 | 1.0000 | 1.0000 |

Name | Notation | Model (Mean, Standard Deviation) |
---|---|---|

Specific impulse stage 1 | ${U}_{Is{p}_{1}}$ | $\mathcal{N}(0,1)$ (additive, $\mathrm{s})$ |

Specific impulse stage 2 | ${U}_{Is{p}_{2}}$ | $\mathcal{N}(0,1)($ additive, $\mathrm{s})$ |

Residual mass stage 1 | ${U}_{{m}_{1}}$ | $\mathcal{U}(-750,750)$ (additive, $\mathrm{kg})$ |

Residual mass stage 2 | ${U}_{{m}_{2}}$ | $\mathcal{U}(-250,250)$ (additive, $\mathrm{kg})$ |

Mass flow rate stage 1 | ${U}_{{q}_{1}}$ | $\mathcal{N}(0,5)$ (additive, $\mathrm{kg}/\mathrm{s})$ |

Mass flow rate stage 2 | ${U}_{{q}_{2}}$ | $\mathcal{N}(0,5)$ (additive, $\mathrm{kg}/\mathrm{s})$ |

Drag coefficient | ${U}_{CD}$ | $\mathcal{U}(-0.05,0.05)$ (additive, −) |

Number of KL Modes | 2 | 5 | 10 | 20 | 40 |
---|---|---|---|---|---|

Q2 | −42.9737 | 0.7874 | 0.9900 | 0.9999 | 1.0000 |

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**MDPI and ACS Style**

Brevault, L.; Balesdent, M.; Valderrama-Zapata, J.-L. Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function. *Appl. Sci.* **2022**, *12*, 10027.
https://doi.org/10.3390/app121910027

**AMA Style**

Brevault L, Balesdent M, Valderrama-Zapata J-L. Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function. *Applied Sciences*. 2022; 12(19):10027.
https://doi.org/10.3390/app121910027

**Chicago/Turabian Style**

Brevault, Loïc, Mathieu Balesdent, and Jorge-Luis Valderrama-Zapata. 2022. "Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function" *Applied Sciences* 12, no. 19: 10027.
https://doi.org/10.3390/app121910027