Wavelet transforms can replace standard backpropagation neurons and radial basis function based neurons to create "wavelet neural nets." These use a wavelet function as their activation function.
"Convolutional wavelet neural networks" take convolution neurons but replace fully connected layers with WNNs.
Wavelet based neural networks train faster (1/th speed of standard backpropagation) and can be slightly more accurate (~2%) than baseline CNNs on the MNIST dataset.
WNN is designed as follows: Firstly, structure of BPNN is adopted as the basic structure of WNN; Secondly, the form of activation function in hidden layers of RBFNN is adopted; Thirdly, the wavelet transform function is adopted as the activation function.