Deep learning-based reduced-order modeling for parameterized convection-dominated partial differential equations

Reduced-order modeling of fluid flows has been an active area of research. It approximates the evolution of physical systems in time in terms of coherent patterns and structures that generally consist of a dimensionality reduction mechanism and a dynamical model in the reduced state space. This paper proposes a deep learning-based reduced-order modeling composed of beta-variational autoencoder, multilayer perceptron, and transformer architectures for problems governed by the parameterized convection-dominated partial differential equations. In our approach, beta-variational autoencoder is utilized as a dimensionality reduction mechanism, transformer is trained to predict the future state of the system, and multilayer perceptron is applied to learn the relationship between different parameter values and latent space representations. Therefore, the future state of the system can be obtained in the online phase. The proposed method is tested on several benchmark convection-dominated partial differential equations, such as Burgers' equation, traffic flow problem, shallow water equation, and Navier-Stokes equation. The results demonstrate the applicability and effectiveness of the proposed reduced-order modeling method for convection-dominated partial differential equations.