CD-ROM: Complemented Deep-Reduced Order Model

TL;DR

This paper introduces CD-ROM, a deep learning closure model for POD-Galerkin reduced order models, improving accuracy and stability in simulating complex nonlinear dynamical systems like fluid flows.

Contribution

It presents a theoretically grounded, interpretable neural network-based closure approach that enhances classical ROMs for nonlinear high-dimensional systems.

Findings

Improved accuracy in fluid dynamics simulations.

Stable and efficient reduced order models.

Successful application to Navier-Stokes and Kuramoto-Sivashinsky equations.

Abstract

Model order reduction through the POD-Galerkin method can lead to dramatic gains in terms of computational efficiency in solving physical problems. However, the applicability of the method to non linear high-dimensional dynamical systems such as the Navier-Stokes equations has been shown to be limited, producing inaccurate and sometimes unstable models. This paper proposes a deep learning based closure modeling approach for classical POD-Galerkin reduced order models (ROM). The proposed approach is theoretically grounded, using neural networks to approximate well studied operators. In contrast with most previous works, the present CD-ROM approach is based on an interpretable continuous memory formulation, derived from simple hypotheses on the behavior of partially observed dynamical systems. The final corrected models can hence be simulated using most classical time stepping schemes. The capabilities of the CD-ROM approach are demonstrated on two classical examples from Computational Fluid Dynamics, as well as a parametric case, the Kuramoto-Sivashinsky equation.