Cauchy activation function and XNet

In today's scientific exploration, the rise of computational technology has marked a significant turning point. Traditional methods of theory and experimentation are now complemented by advanced computational tools that tackle the complexity of real-world systems. Machine learning, particularly deep neural networks, has led to breakthroughs in fields like image processing and language understanding [3, 7], and its application to scientific problems-such as predicting protein structures [9, 10] or forecasting weather [13]-demonstrates its potential to revolutionize our approach. One of the primary challenges in computational mathematics and artificial intelligence (AI) lies in determining the most appropriate function to accurately model a given dataset. In machine learning, the objective is to leverage such functions for predictive purposes. Traditional methods rely on predetermined classes of functions, such as polynomials or Fourier series, which, though simple and computationally manageable, may limit the flexibility and accuracy of the fit. In contrast, modern deep learning neural networks primarily employ locally linear functions with nonlinear activations.