  Machine-learning models have demonstrated a great ability to learn complex
patterns and make predictions. In high-dimensional nonlinear problems of fluid
dynamics, data representation often greatly affects the performance and
interpretability of machine learning algorithms. With the increasing
application of machine learning in fluid dynamics studies, the need for
physically explainable models continues to grow. This paper proposes a feature
learning algorithm based on variational autoencoders, which is able to assign
physical features to some latent variables of the variational autoencoder. In
addition, it is theoretically proved that the remaining latent variables are
independent of the physical features. The proposed algorithm is trained to
include shock wave features in its latent variables for the reconstruction of
supercritical pressure distributions. The reconstruction accuracy and physical
interpretability are also compared with those of other variational
autoencoders. Then, the proposed algorithm is used for the inverse design of
supercritical airfoils, which enables the generation of airfoil geometries
based on physical features rather than the complete pressure distributions. It
also demonstrates the ability to manipulate certain pressure distribution
features of the airfoil without changing the others.
