In this talk, we present a class of optimization-based numerical methods for the simulation of high-speed compressible flows. The proposed framework leverages nonlinear finite element manifolds that adapt the approximation space to the most salient flow features, including boundary layers, shock waves, and rarefaction regions. By embedding problem-specific structure directly into the trial space, these methods enhance accuracy and efficiency, particularly in regimes characterized by strong gradients and discontinuities. Both the linear coefficients and the nonlinear parameters defining the approximation manifold are determined through the solution of an optimization problem. Numerical results demonstrate the potential of this approach to deliver highly accurate solutions on extremely coarse grids.