In this paper, we propose a general numerical framework to derive structure-preserving reduced order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a strategic process to devise accurate, efficient, and structure-preserving numerical algorithms to solve the forehead reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws. The proposed numerical platform is unique from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent one, where the transformed system's free energy adopts a quadratic form of the state variables. This transformation is named energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced order models. The reduced order models derived through our method continue to uphold the energy dissipation law. Secondly, our proposed numerical approach automatically provides numerical algorithms to discretize the reduced order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. In essence, our platform offers a distinctive approach to derive structure-preserving reduced-order models for a wide range of PDE systems abiding by thermodynamic principles.

翻译：本文提出了一种通用数值框架，用于推导热力学一致偏微分方程的结构保持降阶模型。该数值框架具有两大核心特征：(a) 为热力学一致偏微分方程系统提取降阶模型的系统化方法，同时保留其固有的热力学原理；(b) 设计精确、高效且结构保持的数值算法以求解上述降阶模型的策略性流程。该平台的通用性可扩展至多种受内嵌热力学定律支配的偏微分方程系统。所提出的数值平台具有以下独特性：首先，它利用广义昂萨格原理将热力学一致偏微分方程系统转化为等价形式，其中转化后系统的自由能呈现状态变量的二次型形式，该变换称为能量二次化。通过能量二次化，我们获得了推导降阶模型的新视角。通过本方法得到的降阶模型持续满足能量耗散律。其次，所提出的数值方法可自动提供离散降阶模型的数值算法，这些算法始终是线性的、易于实现和求解，且解具有唯一性。此外，这些算法先天性地确保了热力学定律的满足。本质上，本平台为推导遵循热力学原理的各类偏微分方程系统的结构保持降阶模型提供了一种独特方案。