This is the first paper in a sequence of studies in which we introduce a new type of neural networks (NNs) -- wavelet-based neural networks (WBNNs) -- and study their properties and potential for applications. We begin this study with a comparison to the currently existing type of wavelet neural networks (WNNs) and show that WBNNs vastly outperform WNNs. One reason for the vast superiority of WBNNs is their advanced hierarchical tree structure based on biorthonormal multiresolution analysis (MRA). Another reason for this is the implementation of our new idea to incorporate the wavelet tree depth into the neural width of the NN. The separation of the roles of wavelet depth and neural depth provides a conceptually and algorithmically simple but highly efficient methodology for sharp increase in functionality of swarm and deep WBNNs and rapid acceleration of the machine learning process.