In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.

翻译：本文针对双曲守恒律方程，提出了一种基于神经网络的有限差分加权本质无振荡（WENO）格式。我们采用监督学习方法，设计了两种损失函数：一种基于均方误差，另一种基于均方对数误差，并以WENO3-JS权重作为标签。每种损失函数包含两个组成部分：第一部分比较神经网络输出的权重与WENO3-JS权重之间的差异；第二部分使神经网络输出的权重与线性权重相匹配。损失函数的前一项强制神经网络遵循WENO特性，这意味着无需后处理层；而后一项则使格式在间断附近具有更好的性能。在神经网络结构方面，为提升计算效率，我们选用浅层神经网络（SNN），其Delta层由归一化未分割差分构成。与WENO3-JS和WENO3-Z的模拟结果相比，所构建的WENO3-SNN格式在一维算例中表现出更优的结果，在二维算例中也展现出改进的性能。