  In this paper, we introduce Wav-KAN, an innovative neural network
architecture that leverages the Wavelet Kolmogorov-Arnold Networks (Wav-KAN)
framework to enhance interpretability and performance. Traditional multilayer
perceptrons (MLPs) and even recent advancements like Spl-KAN face challenges
related to interpretability, training speed, robustness, computational
efficiency, and performance. Wav-KAN addresses these limitations by
incorporating wavelet functions into the Kolmogorov-Arnold network structure,
enabling the network to capture both high-frequency and low-frequency
components of the input data efficiently. Wavelet-based approximations employ
orthogonal or semi-orthogonal basis and maintain a balance between accurately
representing the underlying data structure and avoiding overfitting to the
noise. While continuous wavelet transform (CWT) has a lot of potentials, we
also employed discrete wavelet transform (DWT) for multiresolution analysis,
which obviated the need for recalculation of the previous steps in finding the
details. Analogous to how water conforms to the shape of its container, Wav-KAN
adapts to the data structure, resulting in enhanced accuracy, faster training
speeds, and increased robustness compared to Spl-KAN and MLPs. Our results
highlight the potential of Wav-KAN as a powerful tool for developing
interpretable and high-performance neural networks, with applications spanning
various fields. This work sets the stage for further exploration and
implementation of Wav-KAN in frameworks such as PyTorch and TensorFlow, aiming
to make wavelets in KAN as widespread as activation functions like ReLU and
sigmoid in universal approximation theory (UAT). The codes to replicate the
simulations are available at https://github.com/zavareh1/Wav-KAN.
