XNet用柯西积分激活函数实现高阶函数逼近，性能远超传统网络。
Enhancing Neural Function Approximation: The XNet Outperforming KAN
- 采用柯西积分构造激活函数，理论支持任意阶多项式收敛
- 函数逼近误差降低最高达50000倍，训练速度提升10倍
- 适合科学计算与AI任务，尤其擅长高精度函数拟合
XNet是一种单层神经网络架构，利用基于柯西积分的激活函数实现高阶函数逼近。通过理论分析，我们证明了XNet中使用的柯西激活函数可实现任意阶多项式收敛，从根本上优于依赖深度增加或B样条激活的传统MLP和科尔莫戈罗夫-阿诺德网络（KAN）。在函数逼近、偏微分方程求解和强化学习的广泛实验中，XNet表现出色——逼近误差最高降低50000倍，训练速度最快提升10倍。这些结果确立了XNet在科学计算与人工智能应用中的高效架构地位。
原文摘要 · Abstract (English)
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.
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