Airfoil optimization is critical to the design of turbine blades and aerial vehicle wings, among other aerodynamic applications. This design process is often constrained by the computational time required to perform CFD simulations on different design options, or the availability of adjoint solvers. A common method to mitigate some of this computational expense in nongradient optimization is to perform dimensionality reduction on the data and optimize the design within this smaller subspace. Although learning these low-dimensional airfoil manifolds often facilitates aerodynamic optimization, these subspaces are often still computationally expensive to explore. Moreover, the complex data organization of many current nonlinear models make it difficult to reduce dimensionality without model retraining. Inducing orderings of latent components restructures the data, reduces dimensionality reduction information loss, and shows promise in providing near-optimal representations in various dimensions while only requiring the model to be trained once. Exploring the response of airfoil manifolds to data and model selection and inducing latent component orderings have potential to expedite airfoil design and optimization processes.

This thesis first investigates airfoil manifolds by testing the performance of linear and nonlinear dimensionality reduction models, examining if optimized geometries occupy lower dimensional manifolds than non-optimized geometries, and by testing if the learned representation can be improved by using target optimization conditions as data set features. We find that autoencoders, although often suffering from stability issues, have increased performance over linear methods such as PCA in low dimensional representations of airfoil databases. We also find that the use of optimized geometry and the addition of performance parameters have little effect on the intrinsic dimensionality of the data. This thesis then explores a recently proposed approach for inducing latent space orderings called Rotation Augmented Gradient (RAG) [1]. We extend their algorithm to nonlinear models to evaluate its efficacy at creating easily-navigable latent spaces with reduced training, increased stability, and improved design space preconditioning. Our extension of the RAG algorithm to nonlinear models has potential to expedite dimensional analyses in cases with near-zero gradients and long training times by eliminating the need to retrain the model for different dimensional subspaces