From Kolmogorov to Cauchy: Shallow XNet Surpasses KANs
Xin Li, Xiaotao Zheng, Zhihong Xia
We study a shallow variant of XNet, a neural architecture with activation functions derived from the Cauchy integral formula. Prior work applied these activations in deep models; we show that even a single-layer XNet achieves near-exponential approximation rates—surpassing the polynomial limits of MLPs and spline-based networks such as Kolmogorov–Arnold Networks (KANs).Empirically, XNet reduces approximation error by over 600× on discontinuous and high-dimensional functions, achieves 10,000× lower residuals in physics-informed PDEs, and improves policy accuracy and sample efficiency on PPO tasks.These results highlight that expressivity can emerge from principled activation design rather than depth. XNet offers a compact, theoretically grounded alternative for approximation, scientific computing, and control.