Enhancing Neural Function Approximation: The XNet Outperforming KAN
This work addresses the efficiency and accuracy challenges in function approximation for scientific computing and AI applications, representing a novel method rather than an incremental improvement.
The paper tackles the problem of neural function approximation by introducing XNet, a single-layer architecture with Cauchy integral-based activation functions, which reduces approximation error by up to 50000 times and accelerates training by up to 10 times compared to existing methods like MLPs and KANs.
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.