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.

翻译：动理学方程对于建模非平衡现象至关重要，但其计算复杂度构成挑战。本文提出一种基于数据驱动的降阶模型（ROM）方法，通过比较本征正交分解（POD）和自编码器神经网络（AE）两种ROM方法，高效模拟动理学方程中的非平衡流动。虽然自编码器初始显示出更高的精度，但随着考虑更多模态，POD的精度得以提升。值得注意的是，我们的工作认识到：经典POD-MOR方法虽能精确表征动理学方程的非线性解流形，但由于数据流形固有的非线性特性，可能无法提供数据的精简模型。我们展示了如何利用自编码器寻找系统的内在维度，并将内在量与具有物理解释的宏观量相关联。