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A common strategy for the dimensionality reduction of nonlinear partialdifferential equations relies on the use of the proper orthogonal decomposition(POD) to identify a reduced subspace and the Galerkin projection for evolvingdynamics in this reduced space. However, advection-dominated PDEs arerepresented poorly by this methodology since the process of truncation discardsimportant interactions between higher-order modes during time evolution. Inthis study, we demonstrate that an encoding using convolutional autoencoders(CAEs) followed by a reduced-space time evolution by recurrent neural networksovercomes this limitation effectively. We demonstrate that a truncated systemof only two latent-space dimensions can reproduce a sharp advecting shockprofile for the viscous Burgers equation with very low viscosities, and asix-dimensional latent space can recreate the evolution of the inviscid shallowwater equations. Additionally, the proposed framework is extended to aparametric reduced-order model by directly embedding parametric informationinto the latent space to detect trends in system evolution. Our results showthat these advection-dominated systems are more amenable to low-dimensionalencoding and time evolution by a CAE and recurrent neural network combinationthan the POD Galerkin technique.

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders