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

arXiv: https://arxiv.org/abs/2009.13415 · HTML: https://mlsanity.com/paper/2009.13415/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2020.

## Built on by (most cited first, 7 of 7)

- [Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure](https://mlsanity.com/paper/2105.05690/) (2021)
- [Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures](https://mlsanity.com/paper/2105.14410/) (2021)
- [Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds](https://mlsanity.com/paper/2109.00700/) (2021)
- [DeePN2: A deep learning-based non-Newtonian hydrodynamic model](https://mlsanity.com/paper/2112.14798/) (2021)
- [Machine learning moment closure models for the radiative transfer equation IV: enforcing symmetrizable hyperbolicity in two dimensions](https://mlsanity.com/paper/2604.20143/) (2026)
- [Data-driven discovery of multiscale chemical reactions governed by the law of mass action](https://mlsanity.com/paper/2101.06589/) (2021)
- [Structure-preserving neural networks for the regularized entropy-based closure of the Boltzmann moment system](https://mlsanity.com/paper/2404.14312/) (2024)

## Abstract (arXiv)

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.