Machine learning (ML) has gained widespread adoption for tackling inverse problems in recent years. However, ensuring the reliability of ML predictions remains a critical challenge, commonly due to a lack of quantitative methods for error estimation.
Since the feature distribution shift between target and training samples is believed to affect model prediction accuracy, this study introduces Fréchet distance (F-value) to quantify the corresponding shift of the model input (aerodynamic distributions) in the airfoil inverse mapping under a convolutional neural network, where the network predicts airfoil geometry from the prescribed aerodynamic distribution.
The analysis reveals that Ftot, the integration of the F-values for a given sample (either pressure coefficient Cp or friction coefficient Cf distribution), exhibits a positive correlation with mean relative error (MRE) in airfoil geometry prediction. Specifically, the expected MRE has a 2.5-fold increase from 0.863% to 2.179% when the Ftot rises from 0.35 to 0.85. These results validate Ftot as an effective error estimation metric in airfoil inverse mapping. Subsequently, this study derives the probability density for achieving an MRE below a specified threshold conditioned on the Ftot Intervals of the testing samples, which can be served as a quantitative foundation for end-to-end error estimation or guidance for optimal training dataset assembly. Furthermore, this paper verifies the applicability of this estimation method in the multi-input (Cp and Cf) model if selecting the smallest Ftot among different feature inputs as the composite Ftot.
The current research enhances the interpretability of prediction errors in neural networks and provides guidance for better feature selection in the multiple feature fusion, improving the reliability of the practical applications.
