Nonlinear Reduced Order Modeling Using Convolution Autoencoder for the Quasi-Geostrophic Equation

Turbulent geophysical flows governed by the quasi-geostrophic equation (QGE) present computational challenges due to their high dimensionality and nonlinear dynamics. Reduced order models (ROMs) aim to address these challenges by reducing the system's complexity. Traditional linear subspace methods, such as proper orthogonal decomposition (POD)-based Galerkin projection, often fail to capture nonlinear phenomena like coherent vortices and small-scale turbulence, leading to stability issues and reduced robustness. This study proposes a nonlinear model reduction framework combining a convolutional autoencoder (CAE) for spatial compression with a neural network-based flow map (FM) to model temporal evolution. Results demonstrate that the CAE-based model (NL-CAE) significantly outperforms the traditional Galerkin-based ROM (G-ROM) at the same reduced dimensionality, offering a more robust and accurate representation of QGE dynamics.