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.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful method for solving both forward and inverse problems involving Partial Differential Equations (PDEs) [1-4]. PINNs leverage the expressive power of neural networks to minimize a loss function that enforces the governing PDEs and boundary/initial conditions. This approach has been widely applied across various domains, including heat transfer [5-7], solid mechanics [8-10], incompressible flows [11-13], stochastic differential equations [14, 15], and uncertainty quantification [16, 17]. Despite their success, PINNs face significant challenges and often struggle to solve certain classes of problems [18, 19]. One major difficulty arises in scenarios where the solution exhibits rapid changes, such as in'stiff' PDEs [20], leading to issues with convergence and accuracy.

In today's scientific exploration, the rise of computational technology has marked a significant turning point. Traditional methods of theory and experimentation are now complemented by advanced computational tools that tackle the complexity of real-world systems. Machine learning, particularly deep neural networks, has led to breakthroughs in fields like image processing and language understanding [3, 7], and its application to scientific problems-such as predicting protein structures [9, 10] or forecasting weather [13]-demonstrates its potential to revolutionize our approach. One of the primary challenges in computational mathematics and artificial intelligence (AI) lies in determining the most appropriate function to accurately model a given dataset. In machine learning, the objective is to leverage such functions for predictive purposes. Traditional methods rely on predetermined classes of functions, such as polynomials or Fourier series, which, though simple and computationally manageable, may limit the flexibility and accuracy of the fit. In contrast, modern deep learning neural networks primarily employ locally linear functions with nonlinear activations.