Neural network-based, structure-preserving entropy closures for the Boltzmann moment system

TL;DR

This paper introduces neural network-based entropy closures for the Boltzmann moment system that preserve the system's structure, enabling faster kinetic solvers with maintained accuracy through convexity and monotonicity constraints.

Contribution

It develops neural network methods that embed convexity and monotonicity to preserve the minimal entropy closure structure in Boltzmann moment systems.

Findings

Neural network closures significantly speed up kinetic solvers.

The methods maintain accuracy comparable to traditional approaches.

Error bounds for neural network generalization are established.

Abstract

This work presents neural network based minimal entropy closures for the moment system of the Boltzmann equation, that preserve the inherent structure of the system of partial differential equations, such as entropy dissipation and hyperbolicity. The described method embeds convexity of the moment to entropy map in the neural network approximation to preserve the structure of the minimal entropy closure. Two techniques are used to implement the methods. The first approach approximates the map between moments and the minimal entropy of the moment system and is convex by design. The second approach approximates the map between moments and Lagrange multipliers of the dual of the minimal entropy optimization problem, which present the gradients of the entropy with respect to the moments, and is enforced to be monotonic by introduction of a penalty function. We derive an error bound for the generalization gap of convex neural networks which are trained in Sobolev norm and use the results to construct data sampling methods for neural network training. Numerical experiments are conducted, which show that neural network-based entropy closures provide a significant speedup for kinetic solvers while maintaining a sufficient level of accuracy. The code for the described implementations can be found in the Github repositories.