# Uncertainty quantification oriented active learning for surrogates of simulators with functional outputs

Lucas Brunel, Mathieu Balesdent, Loïc Brevault, Rodolphe Le Riche, Bruno Sudret

<figure><img src="image_1.png" /></figure>

## To cite this version:

Lucas Brunel, Mathieu Balesdent, Loïc Brevault, Rodolphe Le Riche, Bruno Sudret. Uncertainty quantification oriented active learning for surrogates of simulators with functional outputs. 9th european congress on computational methods in applied sciences and engineering (ECCOMAS 2024), ECCOMAS, Jun 2024, Lisbonne, Portugal. hal-04800967

HAL Id: hal-04800967

https://hal.science/hal-04800967v1

Submitted on 24 Nov 2024

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Liberté
Égalité
Fraternité

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_4.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

# Uncertainty quantification oriented active learning for surrogates of simulators with functional outputs

L. Brunel<sup>1,2</sup>, M. Balesdent<sup>1</sup>, L. Brevault<sup>1</sup>, R. Le Riche<sup>2</sup> & B. Sudret<sup>3</sup>

ECCOMAS, Lisbon, Portugal

- 5 June 2024 -

<sup>1</sup>ONERA, Paris Saclay University, <sup>2</sup>LIMOS (CNRS, Mines Saint-Etienne and Université Clermont Auvergne), <sup>3</sup>ETH Zürich

Ce document est la propriété de l'ONERA. Il ne peut être communiqué à des tiers et/ou reproduit sans l'autorisation préalable écrite de l'ONERA, et son contenu ne peut être divulgué.

This document and the information contained herein is proprietary information of ONERA and shall not be disclosed or reproduced without the prior authorization of ONERA.
* Design of complex systems (e.g., aerospace vehicles) using **costly simulations with functional outputs** (black box functions)
* Design of complex systems (e.g., aerospace vehicles) using **costly simulations with functional outputs** (black box functions)

* **Uncertainty quantification (UQ), i.e.,** quantify the impact of the **uncertain input variables vector u** on the outputs to help decision making (e.g., positioning of thermal protections on an hypersonic vehicle)

<figure><img src="image_5.png" /></figure>
* Design of complex systems (e.g., aerospace vehicles) using **costly simulations with functional outputs** (black box functions)

* Uncertainty quantification (UQ), i.e., quantify the impact of the **uncertain input variables vector u** on the outputs to help decision making (e.g., positioning of thermal protections on an hypersonic vehicle)

Design and
environment
uncertainties

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

Samples of
stochastic fields
# Objectives

* Compute the estimator $\hat{q}_{\alpha}$ of the $\alpha$-quantile $q_{\alpha}$ of the simulator output field $(\alpha \in [0, 1])$
* Compute the estimator $\hat{q}_{\alpha}$ of the $\alpha$-quantile $q_{\alpha}$ of the simulator output field ($\alpha \in [0, 1]$)

* Scalar $\alpha$-quantile:

$$q_{\alpha} = \inf \{y \in \mathbb{R} : \alpha \le \Pr[Y \le y]\}$$

<figure><img src="image_8.png" /><figcaption>Figure: Cumulative density function and 0.9-quantile</figcaption></figure>

<figure><img src="image_9.png" /></figure>
# Objectives

* Compute the estimator $\hat{q}_{\alpha}$ of the
  $\alpha$-quantile $q_{\alpha}$ of the simulator output field
  ($\alpha \in [0, 1]$)

* Scalar $\alpha$-quantile:

$$q_{\alpha} = \inf \{y \in \mathbb{R} : \alpha \le \Pr[Y \le y]\}$$

* Extension to functional outputs: one scalar quantile per node of the mesh $\mathcal{X}$

<figure><img src="image_10.png" /><figcaption>Figure: Cumulative density function and 0.9-quantile</figcaption></figure>
# Objectives

* Compute the estimator $\hat{q}_{\alpha}$ of the $\alpha$-quantile $q_{\alpha}$ of the simulator output field ($\alpha \in [0, 1]$)

* Scalar $\alpha$-quantile:

$$q_{\alpha} = \inf \{y \in \mathbb{R} : \alpha \le \Pr[Y \le y]\}$$

* Extension to functional outputs: one scalar quantile per node of the mesh $\mathcal{X}$

* The estimation requires many runs of the costly simulator $\Rightarrow$ need for **surrogate modeling**

<figure><img src="image_11.png" /><figcaption>Figure: Cumulative density function and 0.9-quantile</figcaption></figure>
# Dimensionality reduction (DR)<fnref n="1" />

* Exploits the spatial/temporal/etc. correlation of the field to describe it with a reduced set of latent variables z

<figure><img src="image_12.png" /></figure>

<footnote n="1">¹L. van der Maaten et al. (2009). Dimensionality Reduction: A Comparative Review. *TICC TR 2009-005*.</footnote>
# Dimensionality reduction (DR)<fnref n="1" />

* Exploits the spatial/temporal/etc. correlation of the field to describe it with a reduced set of **latent variables** z

* To each **snapshot** of the output field corresponds a point in the **low-dimensional latent space**

<figure><img src="image_13.png" /></figure>

<footnote n="1">¹L. van der Maaten et al. (2009). Dimensionality Reduction: A Comparative Review. *TICC TR 2009–005*.</footnote>
# Dimensionality reduction (DR)<fnref n="1" />

* Exploits the spatial/temporal/etc. correlation of the field to describe it with a reduced set of **latent variables** z

* To each **snapshot** of the output field corresponds a point in the **low-dimensional latent space**

* Example (on the right): snapshots are mapped to a 1-dimensional manifold

<figure><img src="image_14.png" /></figure>

<footnote n="1">¹L. van der Maaten et al. (2009). Dimensionality Reduction: A Comparative Review. *TiCC TR 2009-005*.</footnote>
* The mapping between the snapshots y and the latent space is **linear**

$$y(\mathbf{u}, x) = \sum_{i=1}^{N} \phi_i(x) z_i(\mathbf{u})$$

<footnote n="2">²Y. C. Liang et al. (2002). Proper Orthogonal Decomposition And Its Applications — Part I: Theory. *Journal of Sound and Vibration* 252.3.</footnote>
* The mapping between the snapshots y and the latent space is **linear**

$$y(\mathbf{u}, x) = \sum_{i=1}^{N} \phi_i(x) z_i(\mathbf{u})$$

* Example (on the right): snapshots (top) and corresponding scaled PCA basis (bottom) of a space launcher trajectory

<figure><img src="image_15.png" /></figure>

²Y. C. Liang et al. (2002). Proper Orthogonal Decomposition And Its Applications — Part I: Theory. *Journal of Sound and Vibration* 252.3.

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>
# Dimensionality reduction (DR)

## Principal component analysis (PCA)²

* The mapping between the snapshots y and the latent space is **linear**

$$y(\mathbf{u}, x) = \sum_{i=1}^{N} \phi_i(x) z_i(\mathbf{u})$$

* Example (on the right): snapshots (top) and corresponding scaled PCA basis (bottom) of a space launcher trajectory

<figure><img src="image_19.png" /></figure>

²Y. C. Liang et al. (2002). Proper Orthogonal Decomposition And Its Applications — Part I: Theory. *Journal of Sound and Vibration* 252.3.

<figure><img src="image_20.png" /></figure>
* The mapping between the snapshots y and the latent space is **linear**

$$y(\mathbf{u}, x) = \sum_{i=1}^{N} \phi_i(x) z_i(\mathbf{u})$$

* Example (on the right): snapshots (top) and corresponding scaled PCA basis (bottom) of a space launcher trajectory

<figure><img src="image_21.png" /></figure>

²Y. C. Liang et al. (2002). Proper Orthogonal Decomposition And Its Applications — Part I: Theory. *Journal of Sound and Vibration* 252.3.
# Gaussian process regression<fnref n="3" /> in the latent space

* Probabilistic machine learning model

<figure><img src="image_22.png" /><figcaption>Figure: Gaussian process regression</figcaption></figure>

<figure><img src="image_23.png" /></figure>

<footnote n="3">³G. Matheron (1963). *Principles of geostatistics*. *Economic Geology* 58.8.</footnote>
# Gaussian process regression<fnref n="3" /> in the latent space

* Probabilistic machine learning model

* The latent variables are modeled by a **Gaussian process (GP):**

$$ \hat{z}_i(\cdot) \sim GP (\mu_i(\cdot), k_i(\cdot, \cdot)) \quad (1) $$

with $\mu_i(\cdot)$ the mean and $k_i(\cdot, \cdot)$ the covariance function

<figure><img src="image_24.png" /><figcaption>Figure: Gaussian process regression</figcaption></figure>

<footnote n="3">³G. Matheron (1963). Principles of geostatistics. *Economic Geology* 58.8.</footnote>
# Gaussian process regression<fnref n="3" /> in the latent space

* Probabilistic machine learning model

* The latent variables are modeled by a **Gaussian process (GP):**

$$ \hat{z}_i(\cdot) \sim GP (\mu_i(\cdot), k_i(\cdot, \cdot)) \quad (1) $$

with $\mu_i(\cdot)$ the mean and $k_i(\cdot, \cdot)$ the covariance function

* Comes with a “confidence” measure represented by the GP variance $\sigma^2(\mathbf{u}, \mathbf{x})$

<figure><img src="image_25.png" /><figcaption>Figure: Gaussian process regression</figcaption></figure>

<footnote n="3">³G. Matheron (1963). Principles of geostatistics. *Economic Geology* 58.8.</footnote>
# Gaussian process regression<fnref n="3" /> in the latent space

* Probabilistic machine learning model

* The latent variables are modeled by a **Gaussian process (GP):**

$$ \hat{z}_i(\cdot) \sim \mathcal{GP}(\mu_i(\cdot), k_i(\cdot, \cdot)) \quad (1) $$

with $\mu_i(\cdot)$ the mean and $k_i(\cdot, \cdot)$ the covariance function

* Comes with a “confidence” measure represented by the GP variance $\sigma^2(\mathbf{u}, \mathbf{x})$

* Used as the **latent surrogate model (LSM)**

<figure><img src="image_26.png" /><figcaption>Figure: Gaussian process regression</figcaption></figure>

<footnote n="3">³G. Matheron (1963). Principles of geostatistics. *Economic Geology* 58.8.</footnote>
# Surrogate model with functional outputs

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_29.png" /></figure>

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_32.png" /></figure>
<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_35.png" /></figure>

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_36.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_37.png" /></figure>
* The learning function $g(\cdot)$ is a compromise between points that are **close to the estimated quantile $\hat{q}_{\alpha}$** (numerator) and points where the **prediction variance is high** (denominator)$^4$

$$g(\mathbf{u}) = \frac{|\mu(\mathbf{u}) - \hat{q}_{\alpha}|}{\sigma(\mathbf{u})} \qquad (2)$$

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

<footnote n="4">⁴R. Schöbi et al. (2017). Rare Event Estimation Using Polynomial-Chaos Kriging. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering 3.2. Publisher: American Society of Civil Engineers.</footnote>
* The learning function $g(\cdot)$ is a compromise between points that are **close to the estimated quantile $\hat{q}_\alpha$** (numerator) and points where the **prediction variance is high** (denominator)$^4$

$$g(\mathbf{u}) = \frac{|\mu(\mathbf{u}) - \hat{q}_\alpha|}{\sigma(\mathbf{u})} \qquad (2)$$

* The sample that will be added is $\mathbf{u}^*$ such that

$$\mathbf{u}^* = \arg \min_{\mathbf{u}} g(\mathbf{u}) \qquad (3)$$

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<footnote>$^4$R. Schöbi et al. (2017). Rare Event Estimation Using Polynomial-Chaos Kriging. *ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering* 3.2. Publisher: American Society of Civil Engineers.</footnote>
* Extension to functional outputs by taking the **integral of the field over the mesh** $\mathcal{X}^5$

$$g(\mathbf{u}) = \int_{\mathcal{X}} \frac{|\mu(\mathbf{u}, x) - \hat{q}_{\alpha}(x)|}{\sigma(\mathbf{u}, x)} d\mathbf{x} \quad (4)$$

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

<footnote n="5">⁵L. Brevault et al. (2022). Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function. *Applied Sciences* 12.19. Number: 19 Publisher: Multidisciplinary Digital Publishing Institute.</footnote>
* Extension to functional outputs by taking the **integral of the field over the mesh** $\mathcal{X}^5$

$$g(\mathbf{u}) = \int_{\mathcal{X}} \frac{|\mu(\mathbf{u}, x) - \hat{q}_{\alpha}(x)|}{\sigma(\mathbf{u}, x)} d\mathbf{x} \quad (4)$$

* The sample that will be added is $\mathbf{u}^*$ such that

$$\mathbf{u}^* = \arg \min_{\mathbf{u}} g(\mathbf{u}) \quad (5)$$

<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>

<footnote n="5">⁵L. Brevault et al. (2022). Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function. *Applied Sciences* 12.19. Number: 19 Publisher: Multidisciplinary Digital Publishing Institute.</footnote>
# UQ oriented active learning with functional outputs

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_50.png" /></figure>

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_51.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_52.png" /></figure>
* With functional outputs, the quantile might not correspond to any snapshot but rather a **composition of snapshots**
* With functional outputs, the quantile might not correspond to any snapshot but rather a **composition of snapshots**

* To illustrate, we divide the mesh into multiple sections and measure which snapshot is the closest to the quantile for each section

<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_55.png" /></figure>

⇒ the quantile corresponds to a single snapshot, i.e., one point in the input space

<figure><img src="image_56.png" /></figure>

RÉPUBLIQUE

FRANÇAISE

Équité
Présentation

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_57.png" /></figure>

AGENCE

INNOVATION

DÉFENSE

<figure><img src="image_58.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_59.png" /></figure>

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Conseil
États-Étrangers

<figure><img src="image_62.png" /></figure>

<figure><img src="image_63.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_64.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

<figure><img src="image_67.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Conseil
Étudiant

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_70.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_71.png" /></figure>

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Conseil
Étudiant

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_76.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>

⇒ the quantile corresponds to two snapshots, i.e., two points in the input space

<figure><img src="image_79.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_80.png" /></figure>

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_81.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

05/06/2024 L. Brunel et al. UQ oriented infill

<figure><img src="image_82.png" /></figure>

13
# The problem with functional quantiles

<figure><img src="image_83.png" /></figure>

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_86.png" /></figure>

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_87.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_88.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_89.png" /></figure>

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Préférentielle

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_92.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_93.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_94.png" /></figure>

<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_97.png" /></figure>

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_98.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_99.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_100.png" /></figure>

<figure><img src="image_101.png" /></figure>

<figure><img src="image_102.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Préférentielle

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_103.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_104.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_105.png" /></figure>

<figure><img src="image_106.png" /></figure>

<figure><img src="image_107.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_108.png" /></figure>

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_109.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_110.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_111.png" /></figure>

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Préférentielle

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_116.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_117.png" /></figure>

<figure><img src="image_118.png" /></figure>

<figure><img src="image_119.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Préférentielle

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_120.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_121.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /></figure>

<figure><img src="image_124.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Préférentielle

<figure><img src="image_125.png" /></figure>

<figure><img src="image_126.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_127.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_128.png" /></figure>

<figure><img src="image_129.png" /></figure>

<figure><img src="image_130.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

École
Polytechnique

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_131.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_132.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_133.png" /></figure>

<figure><img src="image_134.png" /></figure>

<figure><img src="image_135.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

École
Polytechnique

<figure><img src="image_136.png" /></figure>

<figure><img src="image_137.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_138.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_139.png" /></figure>

<figure><img src="image_140.png" /></figure>

<figure><img src="image_141.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

École
Polytechnique

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_142.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_143.png" /></figure>
# The problem with functional quantiles

<figure><img src="image_144.png" /></figure>

<figure><img src="image_145.png" /></figure>

⇒ the entire input space contributes to the quantile

<figure><img src="image_146.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Conseil
Économique
Préparé

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_147.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_148.png" /></figure>
* Proposed approach compared to a **naive latin hypercube sampling (LHS)** scheme of the maximum budget and an active learning criterion based on the **maximization of the surrogate model variance**

<figure><img src="image_149.png" /></figure>
Application to test cases

* Proposed approach compared to a **naive latin hypercube sampling** (LHS) scheme of the maximum budget and an active learning criterion based on the **maximization of the surrogate model variance**

* 20 repetitions for each test case and each criterion
* Proposed approach compared to a **naive latin hypercube sampling** (LHS) scheme of the maximum budget and an active learning criterion based on the **maximization of the surrogate model variance**

* **20 repetitions** for each test case and each criterion

* The performance is measured through the **mean relative error (MRE)** of the estimated quantile
Application to test cases

* Proposed approach compared to a **naive latin hypercube sampling** (LHS) scheme of the maximum budget and an active learning criterion based on the **maximization of the surrogate model variance**

* **20 repetitions** for each test case and each criterion

* The performance is measured through the **mean relative error** (MRE) of the estimated quantile

* We do not solve the minimization problem (find $u^*$), we rely on a discretization of the input space
Goal: compute the 99%-quantile of

$$f(\mathbf{u}, x) = \tilde{a} \exp \left( -\frac{1}{2} \times \frac{0.5 + 0.01\tilde{a} - x}{\ln(1 + u_1)} \right) \quad (6)$$

with $\tilde{a} = 1 + \sin(2nu_2)$, $n = 3$.

The distribution of the uncertain variables $\mathbf{u}$ are $u_1 \sim U(0.2, 0.4)$ and $u_2 \sim U(0, \pi)$.

The mesh coordinate is $x \in [0, 1]$ with 101 nodes.
<figure><img src="image_150.png" /><figcaption>Figure: Distribution in the input space</figcaption></figure>

<figure><img src="image_151.png" /><figcaption>Figure: Snapshots and quantile</figcaption></figure>

<figure><img src="image_152.png" /></figure>
## With 3 regions that activate the quantile

<figure><img src="image_153.png" /></figure>

<figure><img src="image_154.png" /></figure>

Input space for quantile oriented infill

Input space for variance based infill

<figure><img src="image_155.png" /></figure>
# Parametric test case to illustrate

With 3 regions that activate the quantile

<figure><img src="image_156.png" /></figure>

Figure: Input space for quantile oriented infill

<figure><img src="image_157.png" /></figure>

Figure: Convergence of the MRE

<figure><img src="image_158.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Présenté

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_159.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_160.png" /></figure>
* Two-Stage-To-Orbit rocket (circular at 250 km)

<figure><img src="image_161.png" /></figure>

<figure><img src="image_162.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

<figure><img src="image_163.png" /></figure>

<figure><img src="image_164.png" /></figure>

<figure><img src="image_165.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_166.png" /></figure>
* Two-Stage-To-Orbit rocket (circular at 250 km)

* The solver is the optimization of the trajectory

<figure><img src="image_167.png" /></figure>

<figure><img src="image_168.png" /></figure>
# Trajectory of a space launcher test case

* Two-Stage-To-Orbit rocket (circular at 250 km)

* The solver is the optimization of the trajectory

* $u_1 \sim \mathcal{N}(0, 1)$ is a perturbation of the **specific impulse** of the first stage (measure of the engine performance) and $u_2 \sim \mathcal{N}(0, 0.05)$ is a perturbation of the **pitch over angle**

<figure><img src="image_169.png" /></figure>

<figure><img src="image_170.png" /></figure>
# Trajectory of a space launcher test case

* Two-Stage-To-Orbit rocket (circular at 250 km)

* The solver is the optimization of the trajectory

* $u_1 \sim \mathcal{N}(0, 1)$ is a perturbation of the **specific impulse** of the first stage (measure of the engine performance) and $u_2 \sim \mathcal{N}(0, 0.05)$ is a perturbation of the **pitch over angle**

* 99%-quantile: flight envelope for safety purposes

<figure><img src="image_171.png" /></figure>
# Trajectory of a space launcher test case

<figure><img src="image_172.png" /><figcaption>Figure: Distribution in the input space</figcaption></figure>

<figure><img src="image_173.png" /><figcaption>Figure: Snapshots and quantile</figcaption></figure>

<figure><img src="image_174.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

ÉTABLISHED

<figure><img src="image_175.png" /></figure>

<figure><img src="image_176.png" /></figure>

<figure><img src="image_177.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_178.png" /></figure>
# Trajectory of a space launcher test case

<figure><img src="image_179.png" /></figure>

<figure><img src="image_180.png" /></figure>

⇒ -11% median MRE at cost 10 and -7% at cost 20 over a naive LHS

<figure><img src="image_181.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Éqar
Présenté

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_182.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

05/06/2024 L. Brunel et al. UQ oriented infill

<figure><img src="image_183.png" /></figure>
* The proposed strategy behavior is as expected **adding samples in areas of the inputs space that contribute the most to the quantile** or that bear a large variance, and it appears to be beneficial
# Takeaways

* The proposed strategy behavior is as expected **adding samples in areas of the inputs space that contribute the most to the quantile** or that bear a large variance, and it appears to be beneficial

* We cannot simply transpose scalar active learning strategies to simulators with functional outputs
* The proposed strategy behavior is as expected **adding samples in areas of the inputs space that contribute the most to the quantile** or that bear a large variance, and it appears to be beneficial

* We cannot simply transpose scalar active learning strategies to simulators with functional outputs

* The effectiveness of our method is **highly dependent on which parts of the input space compose the quantile**, considering that the exact quantile is not known a priori

<figure><img src="image_184.png" /></figure>
# Future works

* Apply the method on **other test cases** (of increased dimensionality and complexity), in particular cases with varying quantile structure to better understand the effectiveness of the proposed method

<figure><img src="image_185.png" /></figure>
Future works

* Apply the method on **other test cases** (of increased dimensionality and complexity), in particular cases with varying quantile structure to better understand the effectiveness of the proposed method

* Extend it to **multi-fidelity** simulators to take advantage of the lower computational cost of lower-fidelity simulators
Thank you for listening!
Any questions?

www.onera.fr
# Parametric test case to illustrate

## With 5 regions that activate the quantile

<figure><img src="image_186.png" /></figure>

Initial samples

Quantile location

samples

<figure><img src="image_187.png" /></figure>

<figure><img src="image_188.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Préférentielle

ONERA

THE FRENCH AEROSPACE LAB

AGENCE
INNOVATION
DÉFENSE
Viscous free fall

<figure><img src="image_189.png" /></figure>

<figure><img src="image_190.png" /></figure>

<figure><img src="image_191.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

ÉTABLISSEMENT
DE
COUVERTURE

ONERA

THE FRENCH AEROSPACE LAB

<figure><img src="image_192.png" /></figure>

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_193.png" /></figure>
Viscous free fall

<figure><img src="image_194.png" /></figure>

Initial samples

Quantile location

samples

<figure><img src="image_195.png" /></figure>

<figure><img src="image_196.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Équité
Préférentielle

ONERA

THE FRENCH AEROSPACE LAB

AGENCE
INNOVATION
DÉFENSE

<figure><img src="image_197.png" /></figure>

<figure><img src="image_198.png" /></figure>
* Adds samples where we want them again

* Ecart moins important que sur le cas analytique

* Quand ce sera animé analyser l'ordre dans lequel les points sont ajoutés

* Le critère variance est moins intéressant qu'un LHS naif