用数据驱动方法学习守恒律模型，自动保证守恒与熵耗散。
Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws
- 将熵稳定设计嵌入神经网络训练过程，无需预设离散格式。
- 联合学习数值通量与熵函数，长期模拟中保持稳定性与守恒性。
- 适合需物理一致性建模的流体动力学、激波传播等场景。
我们提出一种神经熵稳定守恒通量形式神经网络（NESCFN），用于直接从解轨迹中学习双曲守恒律及其关联熵函数，无需任何预先定义的数值离散化。尽管现有神经网络架构已成功将经典数值原理融入学习模型，但大多依赖于控制方程先验知识或固定离散化假设。本方法通过将熵稳定设计原则内嵌至学习过程，彻底消除对先验知识的依赖，实现完全数据驱动下的物理一致动态发现。通过联合学习数值通量函数与对应熵函数，该方法确保了守恒性与熵耗散性，这对双曲守恒律系统的长期稳定性和保真度至关重要。数值结果表明，该方法在长时间演化中保持稳定性与守恒性，并准确捕捉激波传播速度，即使训练数据中未提供未来时间解信息。
原文摘要 · Abstract (English)
We propose a neural entropy-stable conservative flux form neural network (NESCFN) for learning hyperbolic conservation laws and their associated entropy functions directly from solution trajectories, without requiring any predefined numerical discretization. While recent neural network architectures have successfully integrated classical numerical principles into learned models, most rely on prior knowledge of the governing equations or assume a fixed discretization. Our approach removes this dependency by embedding entropy-stable design principles into the learning process itself, enabling the discovery of physically consistent dynamics in a fully data-driven setting. By jointly learning both the numerical flux function and a corresponding entropy, the proposed method ensures conservation and entropy dissipation, critical for long-term stability and fidelity in the system of hyperbolic conservation laws. Numerical results demonstrate that the method achieves stability and conservation over extended time horizons and accurately captures shock propagation speeds, even without oracle access to future-time solution profiles in the training data.
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