From Kolmogorov to Cauchy: Shallow XNet Surpasses KANs

Xin Li (Huazhong University of Science and Technology) · Xiaotao Zheng (Soochow University) · Zhihong Xia (Northwestern University)

activation designapproximation ratescauchy integral formulacontroldiscontinuous functionsfunction approximationkolmogorov–arnold networksmlpsphysics-informed pdespolicy accuracypolynomial boundsppo-based reinforcement learningsample efficiencyscientific computingshallow neural architecturexnet

We study a shallow variant of XNet, a neural architecture whose activation functions are derived from the Cauchy integral formula. While prior work focused on deep variants, we show that even a single-layer XNet exhibits near-exponential approximation rates—exceeding the polynomial bounds of MLPs and spline-based networks such as Kolmogorov–Arnold Networks (KANs). Empirically, XNet reduces approximation error by over 600× on discontinuous functions, achieves up to 20,000× lower residuals in physics-informed PDEs, and improves policy accuracy and sample efficiency in PPO-based reinforcement learning—while maintaining comparable or better computational efficiency than KAN baselines. These results demonstrate that expressive approximation can stem from principled activation design rather than depth alone, offering a compact, theoretically grounded alternative for function approximation, scientific computing, and control.