Universal approximation theorem

In the mathematical theory of artificial neural networks, universal approximation theorems are results that put limits on what neural networks can theoretically learn, i.e. that establish the density of an algorithmically generated class of functions within a given function space of interest. Typically, these results concern the approximation capabilities of the feedforward architecture on the space of continuous functions between two Euclidean spaces, and the approximation is with respect to the compact convergence topology. What must be stressed, is that while some functions can be arbitrarily well approximated in a region, the proofs do not apply outside of the region, i.e. the approximated functions do not extrapolate outside of the region. That applies for all non-periodic activation functions, i.e. what's in practice used and most proofs assume. However, there are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis functions, or neural networks with specific properties. Most universal approximation theorems can be parsed into two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("bounded depth and bounded width" case). Universal approximation theorems imply that neural networks can represent a wide variety of interesting functions with appropriate weights. On the other hand, they typically do not provide a construction for the weights, but merely state that such a construction is possible.