Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-equilibrium Flows

TL;DR

This paper introduces a method to learn interpretable, thermodynamically stable, and Galilean invariant PDEs for non-equilibrium flows using neural networks, ensuring physical consistency and accuracy across various conditions.

Contribution

It develops a novel approach to learn PDEs that are thermodynamically stable and Galilean invariant, based on the Conservation-dissipation Formalism, with neural networks ensuring physical principles.

Findings

Learned PDEs achieve good accuracy across different Knudsen numbers.

The method can handle discontinuous initial data and shock tube problems.

PDEs are hyperbolic balance laws satisfying conservation and dissipation principles.

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.