Model Order Reduction for the 1D Boltzmann-BGK Equation: Identifying Intrinsic Variables Using Neural Networks

TL;DR

This paper compares traditional POD and neural network autoencoders for reduced order modeling of the 1D Boltzmann-BGK equation, highlighting how autoencoders can identify intrinsic variables and better capture non-linear solution manifolds.

Contribution

It introduces a data-driven approach using autoencoders to identify intrinsic variables and improve model reduction for kinetic equations, comparing it with classical POD methods.

Findings

Autoencoders initially outperform POD in accuracy.

POD's accuracy improves with more modes.

Autoencoders help identify intrinsic system dimensions.

Abstract

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows in kinetic equations by comparing two ROM approaches: Proper Orthogonal Decomposition (POD) and autoencoder neural networks (AE). While AE initially demonstrate higher accuracy, POD's precision improves as more modes are considered. Notably, our work recognizes that the classical POD-MOR approach, although capable of accurately representing the non-linear solution manifold of the kinetic equation, may not provide a parsimonious model of the data due to the inherently non-linear nature of the data manifold. We demonstrate how AEs are used in finding the intrinsic dimension of a system and to allow correlating the intrinsic quantities with macroscopic quantities that have a physical interpretation.