Reduced Order Model Closures: A Brief Tutorial

Reduced order models (ROMs) are computational models with significantly lower dimensions than full order models (FOMs), offering substantial computational savings. However, under-resolved ROMs, which have insufficient dimensions to capture complex dynamics, produce inaccurate results. This tutorial addresses the need for ROM closure models to enhance accuracy in under-resolved regimes. It begins with a toy problem to illustrate the ROM closure concept and then delves into the construction of classical Galerkin ROMs. The tutorial highlights the ROM closure problem and its solution through the ideal ROM, which includes a closure term to model the effect of unresolved scales. It also introduces data-driven variational multiscale ROM (D2-VMS-ROM) as a practical closure model, demonstrating its effectiveness through numerical results. The chapter concludes with a discussion of the mathematical foundations and future research directions in ROM closures.