The weighted essentially non-oscillatory (WENO) scheme is widely used in fluid mechanics and other numerical simulation fields because of its high precision and low oscillation characteristics when dealing with hyperbolic conservation law equations containing discontinuities and high-gradient regions. However, the calculation of nonlinear weights in the WENO reconstruction process is complex and entails a high computational cost, especially when addressing two-dimensional and higher-dimensional problems, resulting in a limited overall computational efficiency. To improve computational efficiency, this study introduces a novel neural network-enhanced weighted essentially non-oscillatory method, abbreviated as WENO-NN. This method replaces the reconstruction process in the WENO scheme. Specifically, we used a subset of the data generated by the WENO method to train a neural network that approximates the functionality of WENO. This approach significantly improved the computational efficiency while preserving accuracy. Further, we evaluated the performance of the WENO-NN scheme on both one-dimensional, two-dimensional, and three-dimensional test cases, including scenarios involving the interaction of strong shocks and shock-density waves. The results demonstrated that the WENO-NN scheme exhibits good versatility across all benchmark tests and resolutions. Its accuracy is comparable to that of the classic WENO scheme, while its computational efficiency is improved by 3 times.