This thesis presents the Sequentially Optimized Meshfree Approximation (SOMA) method, a new and powerful Computational Fluid Dynamics (CFD) solver. While standard computational methods can be faster and cheaper that physical experimentation, both in cost and work time, these methods do have some time and user interaction overhead which SOMA eliminates. As a meshfree method which could use adaptive domain refinement methods, SOMA avoids the need for user generated and/or analyzed grids, volumes, and meshes. Incremental building of a feed-forward artificial neural network through machine learning to solve the flow problem significantly reduces user interaction and reduces computational cost. This is done by avoiding the creation and inversion of possibly dense block diagonal matrices and by focusing computational work on regions where the flow changes and ignoring regions where no changes occur.