用神经网络替代传统WENO权重计算，提升精度与稳定性。
Conservative approximation-based feedforward neural network for WENO schemes
- 用神经网络根据三点值直接预测非线性权重
- 在多种场景下精度接近WENO5-JS，优于WENO3-Z
- 适合需要高精度解的守恒律方程数值模拟
本文提出一种基于保守近似导数的前馈神经网络，用于求解双曲守恒律的加权本质无振荡（WENO）格式。该神经网络输入为三点模板的点值，输出为两个非线性权重，取代传统WENO权重计算过程。训练阶段采用监督学习，构建新的带标签数据集，通过给定点值构造数值通量函数，使通量差分逼近导数至高阶精度。损失函数引入对称平衡项，促使神经网络同时满足保守近似与对称性要求，该性质与WENO3-JS和WENO3-Z一致。由此得到的WENO3-CADNNs在多种基准测试场景和分辨率下均表现出强泛化能力，性能优于WENO3-Z，精度可媲美WENO5-JS。
原文摘要 · Abstract (English)
In this work, we present the feedforward neural network based on the conservative approximation to the derivative from point values, for the weighted essentially non-oscillatory (WENO) schemes in solving hyperbolic conservation laws. The feedforward neural network, whose inputs are point values from the three-point stencil and outputs are two nonlinear weights, takes the place of the classical WENO weighting procedure. For the training phase, we employ the supervised learning and create a new labeled dataset for one-dimensional conservative approximation, where we construct a numerical flux function from the given point values such that the flux difference approximates the derivative to high-order accuracy. The symmetric-balancing term is introduced for the loss function so that it propels the neural network to match the conservative approximation to the derivative and satisfy the symmetric property that WENO3-JS and WENO3-Z have in common. The consequent WENO schemes, WENO3-CADNNs, demonstrate robust generalization across various benchmark scenarios and resolutions, where they outperform WENO3-Z and achieve accuracy comparable to WENO5-JS.
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