A Gaussian moment method and its augmentation via LSTM recurrent neural networks for the statistics of cavitating bubble populations

Abstract: Phase-averaged dilute bubbly flow models require high-order statistical moments of the bubble population. The method of classes, which directly evolve bins of bubbles in the probability space, are accurate but computationally expensive. Moment-based methods based upon a Gaussian closure present an opportunity to accelerate this approach, particularly when the bubble size distributions are broad (polydisperse). For linear bubble dynamics a Gaussian closure is exact, but for bubbles undergoing large and nonlinear oscillations, it results in a large error from misrepresented higher-order moments. Long short-term memory recurrent neural networks, trained on Monte Carlo truth data, are proposed to improve these model predictions. The networks are used to correct the low-order moment evolution equations and improve prediction of higher-order moments based upon the low-order ones. Results show that the networks can reduce model errors to less than $1\%$ of their unaugmented values.