Kolmogorov-Arnold Networks (KANs) promise higher expressive capability and stronger interpretability than Multi-Layer Perceptron, particularly in the domain of AI for Science. However, practical adoption has been hindered by low GPU utilization of existing parallel implementations. To address this challenge, we present a GPU-accelerated operator library, named PolyKAN which is the first general open-source implementation of KAN and its variants. PolyKAN fuses the forward and backward passes of polynomial KAN layers into a concise set of optimized CUDA kernels. Four orthogonal techniques underpin the design: (i) \emph{lookup-table} with linear interpolation that replaces runtime expensive math-library functions; (ii) \emph{2D tiling} to expose thread-level parallelism with preserving memory locality; (iii) a \emph{two-stage reduction} scheme converting scattered atomic updates into a single controllable merge step; and (iv) \emph{coefficient-layout reordering} yielding unit-stride reads under the tiled schedule. Using a KAN variant, Chebyshev KAN, as a case-study, PolyKAN delivers $1.2$--$10\times$ faster inference and $1.4$--$12\times$ faster training than a Triton + cuBLAS baseline, with identical accuracy on speech, audio-enhancement, and tabular-regression workloads on both highend GPU and consumer-grade GPU.