Nonclosedness of sets of neural networks in Sobolev spaces

We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly m-times differentiable activation function ρ, we construct a sequence of neural networks [Formula: see text] whose realizations converge in order-(m-1) Sobolev norm to a function that cannot be realized exactly by a neural network. Thus, sets of realized neural networks are not closed in order-(m-1) Sobolev spaces W^m-1,p for p∈[1,∞). We further show that these sets are not closed in W^m,p under slightly stronger conditions on the mth derivative of ρ. For a real analytic activation function, we show that sets of realized neural networks are not closed in W^k,p for anyk∈N. The nonclosedness allows for approximation of non-network target functions with unbounded parameter growth. We partially characterize the rate of parameter growth for most activation functions by showing that a specific sequence of realized neural networks can approximate the activation function's derivative with weights increasing inversely proportional to the L^p approximation error. Finally, we present experimental results showing that networks are capable of closely approximating non-network target functions with increasing parameters via training.