Thermodynamically consistent physics-informed neural networks for hyperbolic systems

TL;DR

This paper introduces a thermodynamically consistent physics-informed neural network framework based on a least squares control volume scheme, improving PDE discretization, boundary condition handling, and conservation properties for hyperbolic systems.

Contribution

It proposes a novel neural network approach that integrates finite volume methods, enabling better handling of boundary conditions, conservation, and thermodynamic biases in hyperbolic PDEs.

Findings

Enhanced PDE solver accuracy and stability.

Ability to fit shock hydrodynamics models to molecular data.

Incorporation of thermodynamic biases into neural network models.

Abstract

Physics-informed neural network architectures have emerged as a powerful tool for developing flexible PDE solvers which easily assimilate data, but face challenges related to the PDE discretization underpinning them. By instead adapting a least squares space-time control volume scheme, we circumvent issues particularly related to imposition of boundary conditions and conservation while reducing solution regularity requirements. Additionally, connections to classical finite volume methods allows application of biases toward entropy solutions and total variation diminishing properties. For inverse problems, we may impose further thermodynamic biases, allowing us to fit shock hydrodynamics models to molecular simulation of rarefied gases and metals. The resulting data-driven equations of state may be incorporated into traditional shock hydrodynamics codes.