  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.
