I apologize for my ignorance, but hope that someone would provide some pointers to what I am sure is a reasonably well-developed body of theory. Consider $C^\infty(U,V)$ where $U \subset R^k$ and $V \subset R^l$ - I am reasonably sure that this space can be given the structure of a an infinite dimensional manifold. Suppose now that I single out some well-defined class of functions, each element of which is specified by a finite collection of parameters. In particular, I am interested in feed-forward neural networks of fixed architecture, so that the weights and biases give a parametrization, and I would like to think of training as a curve in this class of functions. Does such a collection have the structure of a non-compact finite-dimensional embedded submanifold, in some topology on $C^\infty(U,V)$? What are some references that might cover such topics? I have had standard graduate courses in differential and Riemannian geometry, but am woefully ignorant in functional analysis. I would be grateful for any pointers to sources that might be relevant.