Development of closures for coarse-scale modeling of multiphase and free surface flows using machine learning.*

The aim of this work is to learn coarse-grained PDEs as well as reduced-order models of those using a data-driven approach. We train a neural network to learn an approximate inertial form: ODEs for the coarse-scale system behavior obtained from the fine-scale simulations of a bubbly multiphase flow in a vertical channel. We average in the direction parallel to the overall flow to create a dataset of one-spatial-dimension, time-dependent profiles. We perform Proper Orthogonal Decomposition (POD) to reduce the high-dimensional averaged snapshot data to a truncated set of 10 leading-mode amplitude coefficients, and further reduce these through an autoencoder. We then train a second neural network to approximate the continuous-time dynamics of the system in terms of the amplitudes of the ``determining'' POD coefficients (after filtering through the autoencoder) and also reconstruct the full solution via a third network that approximates the remaining POD coefficients as a function of the determining ones. Finally, we also learn a ``grey-box'' model for the right-hand-side operator of the averaged PDE that uses the known parts. To evolve the relevant fields, a pair of unknown closure terms, the wall-normal liquid flux, and summed dissipative terms are learned from coarse evolution data, using only spatial local information.