Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-equilibrium Flows
This work addresses the challenge of modeling non-equilibrium flows with physically consistent PDEs, which is incremental as it builds on existing conservation-dissipation formalism and neural network parameterization.
The authors tackled the problem of learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) for non-equilibrium flows, achieving good accuracy across a wide range of Knudsen numbers and generalizing well to discontinuous initial data and shock tube problems despite training only on smooth data.
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.