An optimization methodology based on neural networks was developed for use in 2D optimal shape design problems. Neural networks were used as a parameterization scheme to represent the shape function, and an edge-based high-resolution scheme for the solution of the compressible Euler equations was used to model the flow around the shape. The global system incorporates neural networks and the Euler fluid solver into the C++ Flood optimization framework containing a library of optimization algorithms. The optimization scheme was applied to a minimal drag problem in an unconstrained optimization case and a constrained case in hypersonic flow using evolutionary training algorithms. The results indicate that the minimum drag problem is solved to a high degree of accuracy but at high computational cost. For more complex shapes, parallel computing methods are required to reduce computational time.