MAGPI: Multifidelity-Augmented Gaussian Process Inputs for Surrogate Modeling from Scarce Data

Abstract

Supervised machine learning describes the practice of fitting a parameterized model to labeled input-output data. Supervised machine learning methods have demonstrated promise in learning efficient surrogate models that can (partially) replace expensive high-fidelity models, making many-query analyses, such as optimization, uncertainty quantification, and inference, tractable. However, when training data must be obtained through the evaluation of an expensive model or experiment, the amount of training data that can be obtained is often limited, which can make learned surrogate models unreliable. However, in many engineering and scientific settings, cheaper \emph{low-fidelity} models may be available, for example arising from simplified physics modeling or coarse grids. These models may be used to generate additional low-fidelity training data. The goal of \emph{multifidelity} machine learning is to use both high- and low-fidelity training data to learn a surrogate model which is cheaper to evaluate than the high-fidelity model, but more accurate than any available low-fidelity model. This work proposes a new multifidelity training approach for Gaussian process regression which uses low-fidelity data to define additional features that augment the input space of the learned model. The approach unites desirable properties from two separate classes of existing multifidelity GPR approaches, cokriging and autoregressive estimators. Numerical experiments on several test problems demonstrate both increased predictive accuracy and reduced computational cost relative to the state of the art.

Figures (6)

• Figure 1: Results from trained models on the analytical test problem. The target functions are plotted with a black dashed line. (top left) The proposed model predictions, (top right) KOH model predictions, (bottom left) NARGP model predictions, (bottom right) single-fidelity kriging model predictions. The shaded regions represent $\pm 2 \sigma$ confidence intervals derived from the Gaussian posteriors of each estimator.
• Figure 2: A visualization of the high-fidelity (USC II) training and testing data. The blue dots show the high-fidelity training data simulated at temperatures 450K and 550K. The black stars show the unseen testing data at temperatures 650K, 750K, and 850K. We emphasize how the behavior of the flame speed outside the design space (temperatures $\leq 550K$) differs significantly from the training data.
• Figure 3: The laminar flame speed experiment captured at four different temperatures. The shaded regions represents a $\pm 2 \sigma$ confidence interval for the predictive posterior of each GP.
• Figure 4: The full high-fidelity (125 $\mu$m LES) flow field with sparse high-fidelity training data indicated with the "+" symbols.
• Figure 5: Results of the sparse flow-field interpolation experiment. The left half of the plots show approximations of high- and low-fidelity flow-fields. The right half of plots shows the error between the approximation to the target (obtained by subtracting the true simulation from the approximation). The first row compares the best low-fidelity simulation to the high-fidelity. The second row compares the KNN approximation of the best low-fidelity simulation to the true best low-fidelity simulation. The third row compares the proposed method's predictions with the true high-fidelity simulation. The last row compares single-fidelity kriging predictions with the true high-fidelity simulation.

Theorems & Definitions (3)

• Proposition 1
• proof
• Remark 1