(542b) A Machine Learning Approach to Bridge the Gap between the Kuramoto-Sivashinsky and the Navier-Stokes Equations for Thin Film Flow

Amplitude equations for the fluid interface are reduced representations of the dominant physics in thin film flows that are typically derived (and valid) under certain limiting conditions. The Kuramoto-Sivashinsky (KS) equation being one of them, is generally considered the first in a hierarchy of one- and two-equation models for the interface evolution of a thin film [1]. The most detailed description of the physics is given by the Navier-Stokes equations, which requires more involved computations and entails increased computational cost.

We present a machine learning approach to leverage an ensemble of data (velocity distributions and amplitude) collected from a CFD model of a thin film flow under conditions where the KS is approximately valid. We use this dataset to train an Artificial Neural Network (ANN) to “learn” the right-hand-side of a PDE describing the time-evolution of the thin film interface. This is then used as part of a time-integrator to capture the dynamics of the interface as a “black box” data-driven amplitude model. We explore further by proposing two cases of “gray”-box models according to which, the “learned” quantity is not the amplitude but two different corrections to the KS prediction [2].

We also exploit the fact that the dataset, which includes the fluid velocity distribution and the interface shape in various Re numbers, can be described by a small number of latent variables. These are derived using both linear (Proper orthogonal decomposition-POD) and nonlinear (Diffusion Maps-DMAPS and autoencoders) [3] methods which determines the “shape” of the approximation of the manifold that contains the data: POD requires several linear hyperplanes to describe a nonlinear manifold, whereas DMAPS delivers a parsimonious representation, requiring less coordinates. We use interpolation methods in the latent space, such as Gappy POD [4] in the linear case and Geometric Harmonics in the nonlinear case, to predict fluid velocity distributions from only a handful of measurements of the interface amplitude. In collaboration with Prof. C. Siettos and Dr. G. Fabiani of the University Federico II, Naples, Italy, we also explore constructing bifurcation diagrams from spatio-temporal observations through Random Projection Neural Networks (RPNNs) [5].

[1] Chang, H. (1994). Wave evolution on a falling film. Annual review of fluid mechanics, 26(1), 103-136.

[2] Lee, S., Siettos, C., Bertalan, T., Amchin, D.B., Bhattacharjee, T., Datta, S., Kevrekidis, I.G. (2020). On the data-driven discovery and calibration of closures. 2020 Virtual AIChE Annual Meeting.

[3] Koronaki, E., Boudouvis, A.G., Evangelou, N., Psarellis, G., Dietrich, F., Kevrekidis, I.G. (2020). From Partial Data to out-of-Bounds Parameter and Observation Estimation with Diffusion Maps and Geometric Harmonics. 2020 Virtual AIChE Annual Meeting.

[4] Willcox, K. (2006). Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Computers & fluids, 35(2), 208-226.

[5] Galaris, E., Fabiani, G., Gallos, I., Kevrekidis, I., & Siettos, C. (2022). Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach. arXiv preprint arXiv:2201.13323.