  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.
