Unified Approximation Theorem for Neural Networks

For any ( f \in \mathcal{F}(\mathbb{R}^n) ) and any ( \epsilon > 0 ), there exists a neural network ( \mathcal{N}(\mathbf{x}; \theta) ) with parameters ( \theta ) such that: [ \sup_{\mathbf{x} \in K} \left| f(\mathbf{x}) - \mathcal{N}(\mathbf{x}; \theta) \right| < \epsilon, ] where ( K \subset \mathbb{R}^n ) is compact.