Deep Reduced-Order Modeling for Fluids: The Importance of Autoencoder Initialization and Initial Transient Data

Many fluids systems may be approximated by dynamics on an inertial manifold, and we wish to obtain these dynamics from data by training a collection of neural networks. In particular, we train an autoencoder to learn the invariant manifold, and learn the dynamics separately, either by training a separate neural network, or a polynomial approximation. Many previous studies limit training data to consider only points that have already relaxed onto the inertial manifold, and neglect the ``transient.'' We find that including the transients can be important, especially for learning projections of initial conditions that are even slightly off the inertial manifold. Furthermore, we find that training time is dramatically decreased if the autoencoder is structured to explicitly include a linear approximation of the inertial manifold. As examples, we construct two-dimensional and eight-dimensional reduced-order models for the complex Ginzburg-Landau equation in its supercritical regime and the Kuramoto-Sivashinsky equation in its chaotic regime, respectively.