Abstract
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.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12071246
Funding statement: This work was supported by the National Natural Science Foundation of China (Grant No. 12071246).
Acknowledgment
JH would like to thank Qi Tang in Los Alamos National Laboratory for helpful discussions in training of the neural networks.
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