From Navier-Stokes simulations for thin films to amplitude equations and back via physics-assisted machine-learning

Amplitude equations for the interfaces in thin film flows are reduced representations of the physics under limiting conditions. The Kuramoto-Sivashinsky (KS) equation is the first in a hierarchy of models, with the most detailed being Navier-Stokes but requiring involved computations. We present a machine learning approach to leverage velocity and amplitude data at various Re from simulations of a thin film flow under conditions where the KS is approximately valid. We use this dataset to train a Neural Network (NN) that learns the PDE describing the time-evolution of the interface, integrated in time as a “black box” to compute the amplitude. We adopt a “gray box” approach to learn a correction to the KS. We also show that local values of the right-hand side of the KS and corresponding derivatives in space and time as input in a NN can predict the dynamics. The approximate nonlinear manifold of the dataset can be parametrized by a small number of latent variables using linear (Proper orthogonal decomposition-POD) and nonlinear (Diffusion Maps and autoencoders) methods. We can then predict fluid velocity distributions from some amplitude data points by interpolating in the latent space with Gappy POD and Geometric Harmonics with the linear and nonlinear methods, respectively.