A Periodic Mapping Activation Function: Mathematical Properties and Application in Convolutional Neural Networks

Activation functions play a crucial role in ensuring training stability, convergence speed, and overall performance in both convolutional and attention-based networks. In this study, we introduce two novel activation functions, each incorporating a sine component and a constraint term. To assess their effectiveness, we replace the activation functions in four representative architectures—VGG16, ResNet50, DenseNet121, and Vision Transformers—covering a spectrum from lightweight to high-capacity models. We conduct extensive evaluations on four benchmark datasets (CIFAR-10, CIFAR-100, MNIST, and Fashion-MNIST), comparing our methods against seven widely used activation functions. The results consistently demonstrate that our proposed functions achieve superior performance across all tested models and datasets. From a design application perspective, the proposed functional periodic structure also facilitates rich and structurally stable activation visualizations, enabling designers to trace model attention, detect surface biases early, and make informed aesthetic or accessibility decisions during interface prototyping.