Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning

We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: a nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. To show the variety of problems the ROM can handle, the methodology is demonstrated on the advection equation, and the flow past a cylinder problem modeled by the incompressible Navier–Stokes equations.