With the increased occurrence of extreme events and miniaturization of aircraft, it has become an urgent task to understand aerodynamics in highly turbulent flight environments. We propose a physics-embedded autoencoder to discover a low-dimensional compact manifold representation of extreme aerodynamics. The present method is demonstrated with the highly nonlinear dynamics of vortex gust-airfoil wake interaction around a NACA0012 airfoil over a range of configurations. The present model extracts key features of the high-dimensional airfoil wake dynamics on a physically interpretable and compact manifold, covering a massive number of wake scenarios across a huge parameter space that determines the characteristics of complex gusty flow conditions. Our data-driven approach offers a new avenue for expressing the seemingly high-dimensional fluid flow systems by identifying the low-dimensional data coordinates that can also be leveraged for data compression and flow control.