In this paper, we introduce an improved version of the fifth-order weighted essentially non-oscillatory (WENO) shock-capturing scheme by incorporating deep learning techniques. The established WENO algorithm is improved by training a compact neural network to adjust the smoothness indicators within the WENO scheme. This modification enhances the accuracy of the numerical results, particularly near abrupt shocks. Unlike previous deep learning-based methods, no additional post-processing steps are necessary for maintaining consistency. We demonstrate the superiority of our new approach using several examples from the literature for the two-dimensional Euler equations of gas dynamics. Through intensive study of these test problems, which involve various shocks and rarefaction waves, the new technique is shown to outperform traditional fifth-order WENO schemes, especially in cases where the numerical solutions exhibit excessive diffusion or overshoot around shocks.

翻译：本文提出了一种改进的五阶加权本质无振荡（WENO）激波捕捉格式，通过引入深度学习技术实现。我们通过训练紧凑型神经网络调整WENO格式中的平滑指示器，改进了经典的WENO算法。该改进提升了数值结果在急剧激波附近的精度。与先前基于深度学习的方法不同，本方法无需额外的后处理步骤来维持一致性。通过文献中多个二维气体动力学欧拉方程算例的测试，我们验证了新方法的优越性。对涉及多种激波和稀疏波的算例进行深入研究表明，该新技术优于传统五阶WENO格式，尤其是在数值解在激波附近出现过扩散或过冲的情况下表现更为突出。