Parallel Layer Normalization for Universal Approximation

Layer Normalization (LN) is not only an optimization component: between affine layers it also supplies nonlinearity, and grouping changes how this nonlinearity scales with width. We study this mechanism through universal approximation. In Parallel Layer Normalization networks (PLN-Nets), fixed-dimensional LN units are applied independently to groups between two affine layers. We prove that a shallow LN-Net with one LN unit is non-universal regardless of width, whereas PLN-Nets become universal as the number of parallel groups grows. The mechanism extends to deep networks, Root Mean Square Normalization (RMSNorm) variants, position-wise feed-forward networks, and convolutional networks under suitable grouping structures. Controlled MLP and physics-informed neural network (PINN) experiments support PLN's practical viability, while CNN and ViT experiments show that group count shapes the finite-sample fitting envelope but does not act alone: the grouped axes, replacement site, and label structure lead to distinct fitting and generalization behavior.