# Data-Driven Correction Reduced Order Models for the Quasi-Geostrophic Equations: A Numerical Investigation

## TL;DR

This study evaluates a data-driven correction reduced order model (DDC-ROM) for simulating quasi-geostrophic equations, demonstrating improved accuracy and stability over traditional methods, especially with constrained variants over extended time intervals.

## Contribution

Introduces and tests a data-driven correction ROM with physical constraints, enhancing simulation accuracy and stability for quasi-geostrophic equations.

## Key findings

- DDC-ROM outperforms standard G-ROM in accuracy.
- Constrained DDC-ROM further improves stability.
- Both models perform well over longer simulation times.

## Abstract

This paper investigates the recently introduced data-driven correction reduced order model (DDC-ROM) in the numerical simulation of the quasi-geostrophic equations. The DDC-ROM uses available data to model the correction term that is generally used to represent the missing information in low-dimensional ROMs. Physical constraints are added to the DDC-ROM to create the constrained data-driven correction reduced order model (CDDC-ROM) in order to further improve its accuracy and stability. Finally, the DDC-ROM is tested on time intervals that are longer than the time interval over which it was trained. The numerical investigation shows that, for low-dimensional ROMs, both the DDC-ROM and CDDC-ROM perform better than the standard Galerkin ROM (G-ROM) and the CDDC-ROM provides the best results.