Goal-Driven Adaptive Sampling Strategies for Machine Learning Models Predicting Fields

TL;DR

This work addresses the challenge of efficiently learning field-predicting surrogates with limited high-fidelity CFD data by introducing a model-agnostic, goal-driven adaptive sampling framework that jointly optimizes scalar QoIs and distributed fields. The method combines Gaussian process surrogates for scalars with field surrogates based on PODI or graph neural networks, and introduces coupled infill criteria (SEwMisfit and JSD) to selectively acquire new simulations that reduce epistemic uncertainty across both output types. Applied to the NASA CRM-NLF case with uncertainty propagation, the coupled strategies consistently outperform scalar-only infill, achieving high accuracy at substantially reduced computational cost and providing reliable uncertainty quantification for both global and field outputs. The approach is broadly applicable to other fields and model architectures, enabling efficient generation of field-aware, uncertainty-aware databases for design, reliability, and optimization tasks.

Abstract

Machine learning models are widely regarded as a way forward to tackle multi-query challenges that arise once expensive black-box simulations such as computational fluid dynamics are investigated. However, ensuring the desired level of accuracy for a certain task at minimal computational cost, e.g. as few black-box samples as possible, remains a challenges. Active learning strategies are used for scalar quantities to overcome this challenges and different so-called infill criteria exists and are commonly employed in several scenarios. Even though needed in various field an extension of active learning strategies towards field predictions is still lacking or limited to very specific scenarios and/or model types. In this paper we propose an active learning strategy for machine learning models that are capable if predicting field which is agnostic to the model architecture itself. For doing so, we combine a well-established Gaussian process model for a scalar reference value and simultaneously aim at reducing the epistemic model error and the difference between scalar and field predictions. Different specific forms of the above-mentioned approach are introduced and compared to each other as well as only scalar-valued based infill. Results are presented for the NASA common research model for an uncertainty propagation task showcasing high level of accuracy at significantly smaller cost compared to an approach without active learning.

Figures (14)

• Figure 1: Coupled goal-driven scalar-field adaptive sampling framework. A Gaussian process (GP) surrogate maps the inputs $\bm{\xi}$ to scalar coefficients resulting in a mean and variance approximation denoted by $\mu_{GP}$ and $\sigma_{GP}$, while a field model receives the same inputs together with other additional information such as the surface-mesh graph and predicts field quantities $F^*$ from which the scalars are integrated out with thereby estimated scalar mean and variance denoted by $\mu_{GNN}$ and $\sigma_{GNN}$. The misfit and epistemic uncertainties of both surrogates feed a coupled infill criterion that selects the next design point. The new snapshot closes the loop and updates both surrogates.
• Figure 2: Conceptual comparison of the four adaptive-sampling criteria used for the coupled GP–GNN surrogate system. The solid lines represent the GP (light-blue) and GNN (dark-blue) predictions of the drag coefficient $C_D(x)$; the shaded bands indicate their epistemic uncertainty. The gray bell curve is the joint input probability density PDF$(x)$. Vertical dashed lines mark candidate infill locations $x_i^*$ proposed by each strategy: $x_1^*$ - Surrogate Error (SE) criterion applied to the GP; $x_2^*$ - SE applied to the GNN; $x_3^*$ - SE with GP–GNN misfit (SEwMisfit); $x_4^*$ - Jensen–Shannon divergence (JSD) coupling the two surrogates. The possibility of almost equally probable next infill sample is represented as $x_{i^{'}}^*$.
• Figure 3: Computational mesh for the NASA CRM.
• Figure 4: (Left) $N_{TS}$ and $Tu$ uncertainty distributions. (Right) Uncertainty distributions for angle of attack, Reynolds number, and Mach number.
• Figure 5: Convergence of the no-infill surrogates. Top: r2-score for $C_L$ (left) and $C_D$ (right); the dashed line marks the 0.99 target. Bottom: normalized RMSE for $C_L$ (left) and $C_D$ (right); the dashed line marks the 3% target. Kriging, PODI and GNN are trained on DoE dataset with size ranging from 10-80 samples.