Exploration of Autoencoder Loss Functions for Reduced-order Modeling of Fluid Flow Data

Data-driven reduced-order models (ROMs) have seen many practical applications in fluid mechanics to reduce the dimensionality of a problem. Fluid flows are notoriously difficult to reduce due to the need to capture phenomena that occur at order-of-magnitude varying scales. An autoencoder is a neural network ROM that can capture non-linear behaviours. In previous literature, the mean-squared error (MSE) has been the prominent loss function in the autoencoder algorithm. However, for non-normalized loss functions such as the MSE, errors in large-scale fluctuations can mask out smaller-scale ones, even though both are equally important to resolve for an accurate representation of the flow field. Furthermore, the MSE is a scalar-based loss while the velocity field is a vector quantity. The use of normalized and vector-based losses will be explored in autoencoder networks to produce a more accurate ROM.

A case study on the incompressible flow past a NACA0012 airfoil at angles of attack from 0^o to 10^o is conducted. The autoencoder loss function is studied in detail, and an alternative to the MSE is proposed which also incorporates conservation principles. Locations on the airfoil surface and in the wake region will be probed to show a more complete picture of the reconstruction error.