Model Comparisons: XNet Outperforms KAN

We initially proposed a novel method for constructing real networks from the complex domain using the Cauchy integral formula in Li et al. (2024); Zhang et al. (2024), utilizing Cauchy kernels as basis functions. This work comprehensively compares these networks with KANs, which use B-spline as basis functions in Liu et al. (2024), and MLPs to highlight our significant improvements. Multi-layer perceptrons (MLPs) (Haykin (1994); Cybenko (1989); Hornik et al. (1989)), recognized as fundamental building blocks in deep learning, have their limitations despite their wide use, particularly in its accuracy, and large number of parameters needed in structures such as in transformers (Vaswani et al. (2017)), and lack interpretability without post-analysis tools (Cunningham et al. (2023)). The Kolmogorov-Arnold Networks (KANs) were introduced as a potential alternative, drawing on the Kolmogorov-Arnold representation theorem (Kolmogorov (1956); Braun & Griebel (2009)), and demonstrate their efficiency and accuracy in computational tasks, especially in solving PDEs and function approximation (Sprecher & Draghici (2002); Köppen (2002); Lin & Unbehauen (1993); Lai & Shen (2021); Leni et al. (2013); Fakhoury et al. (2022)). In the swiftly advancing domain of deep learning, the continuous search for novel neural network designs that deliver superior accuracy and efficiency is pivotal. While traditional activation functions such as the Rectified Linear Unit (ReLU) (Nair & Hinton (2010)) have been widely adopted due to their straightforwardness and efficacy in diverse applications, their shortcomings become evident as the complexity of challenges escalates. This is particularly true in areas that demand meticulous data fitting and the solutions of intricate partial differential equations (PDEs).