Bypassing quadrature moment method instability via recurrent neural networks with application to cavitating bubble dispersions*

Numerical models of disperse bubbly, cavitating flow require high-order moments of the dispersion statistics. The quadrature method of moments (QMOM) provide a framework for approximating these moments. QMOM carries a finite set of raw statistical moments and inverts them for an optimal quadrature rule as needed. However, moment-set realizability and moment-transport-equation instability can prohibit extending to arbitrarily high-order moments, thus limiting attainable accuracy. This limit is encountered when modeling a cavitating bubble population via conditional QMOM (CQMOM). For example, we show that even three-node CQMOM closure of the Rayleigh--Plesset equation is unstable (though two-node is stable). We treat this issue by dynamically altering the two-node quadrature rule via a long short-term memory recurrent neural network. The network is trained on Monte Carlo data and utilizes a novel loss function that penalizes both the error in the computed moments and unrealizable features in the projected moment set. This approach reduces the relative error of the high-order moments by about a factor of ten without numerical instabilities. Further improvement is seen when augmenting the quadrature rule with additional quadrature points.