Using neural networks to accelerate the solution of the Boltzmann equation

TL;DR

This paper introduces a neural network-based approach to accelerate Boltzmann equation simulations while preserving key physical properties like conservation and fluid limits, combining classical modeling with machine learning for improved efficiency.

Contribution

It presents a hybrid neural network method that guarantees conservation and correct fluid dynamic limits, integrating physics-based modeling with deep learning for Boltzmann equation solutions.

Findings

The method preserves conservation properties.

It achieves efficient training and accurate results.

Numerical tests validate multi-scale applicability.

Abstract

One of the biggest challenges for simulating the Boltzmann equation is the evaluation of fivefold collision integral. Given the recent successes of deep learning and the availability of efficient tools, it is an obvious idea to try to substitute the evaluation of the collision operator by the evaluation of a neural network. However, it is unlcear whether this preserves key properties of the Boltzmann equation, such as conservation, invariances, the H-theorem, and fluid-dynamic limits. In this paper, we present an approach that guarantees the conservation properties and the correct fluid dynamic limit at leading order. The concept originates from a recently developed scientific machine learning strategy which has been named "universal differential equations". It proposes a hybridization that fuses the deep physical insights from classical Boltzmann modeling and the desirable computational efficiency from neural network surrogates. The construction of the method and the training strategy are demonstrated in detail. We conduct an asymptotic analysis and illustrate its multi-scale applicability. The numerical algorithm for solving the neural network-enhanced Boltzmann equation is presented as well. Several numerical test cases are investigated. The results of numerical experiments show that the time-series modeling strategy enjoys the training efficiency on this supervised learning task.