Enhancing Neural Function Approximation: The XNet Outperforming KAN

TL;DR

XNet is a novel single-layer neural network using Cauchy-based activations, achieving superior high-order function approximation and outperforming traditional deep models in accuracy and training speed.

Contribution

The paper introduces XNet, a single-layer neural network with Cauchy integral activations, offering arbitrary-order polynomial convergence and outperforming existing architectures like KAN.

Findings

XNet reduces approximation error by up to 50,000 times.

XNet accelerates training by up to 10 times.

XNet outperforms traditional MLPs and KANs in various tasks.

Abstract

XNet is a single-layer neural network architecture that leverages Cauchy integral-based activation functions for high-order function approximation. Through theoretical analysis, we show that the Cauchy activation functions used in XNet can achieve arbitrary-order polynomial convergence, fundamentally outperforming traditional MLPs and Kolmogorov-Arnold Networks (KANs) that rely on increased depth or B-spline activations. Our extensive experiments on function approximation, PDE solving, and reinforcement learning demonstrate XNet's superior performance - reducing approximation error by up to 50000 times and accelerating training by up to 10 times compared to existing approaches. These results establish XNet as a highly efficient architecture for both scientific computing and AI applications.