Model order reduction for the 1D Boltzmann-BGK equation: identifying intrinsic variables using neural networks

• DOI: 10.1007/s10404-024-02711-5
• Publication Date: 2024-02-28T12:02:51Z
• Authors: Julian Koellermeier; Philipp Krah; Julius Reiss; Zachary Schellin
• Keywords: Neural autoencoder networks; Data-driven methods; Kinetic equations; Model order reduction; Fluid Dynamics (physics.flu-dyn); FOS: Mathematics; Boltzmann-BGK; Sod shock tube; FOS: Physical sciences; Physics - Fluid Dynamics; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); Proper orthogonal decomposition

AbstractKinetic 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 model order reduction 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.