In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the conservation-dissipation formalism of irreversible thermodynamics. As governing equations for non-equilibrium flows in one dimension, the learned PDEs are parameterized by fully connected neural networks and satisfy the conservation-dissipation principle automatically. In particular, they are hyperbolic balance laws and Galilean invariant. The training data are generated from a kinetic model with smooth initial data. Numerical results indicate that the learned PDEs can achieve good accuracy in a wide range of Knudsen numbers. Remarkably, the learned dynamics can give satisfactory results with randomly sampled discontinuous initial data and Sod’s shock tube problem although it is trained only with smooth initial data.

中文翻译：

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学习非平衡流动的热力学稳定和伽利略不变偏微分方程

在这项工作中，我们开发了一种基于不可逆热力学的守恒耗散形式来学习可解释、热力学稳定和伽利略不变偏微分方程 (PDE) 的方法。作为一维非平衡流动的控制方程，学习到的偏微分方程由全连接神经网络参数化，并自动满足守恒耗散原理。特别是，它们是双曲平衡定律和伽利略不变量。训练数据是从具有平滑初始数据的动力学模型生成的。数值结果表明，学习到的偏微分方程可以在很宽的克努森数范围内实现良好的准确性。值得注意的是，