Universal Approximation Theorem

The Universal Approximation Theorem states that a feedforward neural network with a single hidden layer containing a finite number of neurons can approximate any continuous function on compact subsets of R^n, given appropriate activation functions and sufficient parameters. Formally, for any continuous function f and any ε > 0, there exists a neural network such that the output of the network is within ε of f for all inputs in the compact set. This theorem underscores the theoretical foundation of neural networks as universal function approximators and has significant implications for the design and application of neural networks in various domains. The theorem is often discussed in the context of approximation theory and highlights the importance of network architecture and activation functions in achieving desired approximation properties.