Data from Hybrid quadrature moment method for accurate and stable representation of non-Gaussian processes applied to bubble dynamics

Solving the population balance equation (PBE) for the dynamics of a dispersed phase coupled to a continuous fluid is expensive. Still, one can reduce the cost by representing the evolving particle density function in terms of its moments. In particular, quadrature-based moment methods (QBMMs) invert these moments with a quadrature rule, approximating the required statistics. QBMMs have been shown to accurately model sprays and soot with a relatively compact set of moments. However, significantly non-Gaussian processes such as bubble dynamics lead to numerical instabilities when extending their moment sets accordingly. We solve this problem by training a recurrent neural network (RNN) that adjusts the QBMM quadrature to evaluate unclosed moments with higher accuracy. The proposed method is tested on a simple model of bubbles oscillating in response to a temporally fluctuating pressure field. The approach decreases model-form error by a factor of 10 when compared with traditional QBMMs. It is both numerically stable and computationally efficient since it does not expand the baseline moment set. Additional quadrature points are also assessed, optimally placed and weighted according to an additional RNN. These points further decrease the error at low cost since the moment set is again unchanged.This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

求解耦合于连续流体的分散相动力学的群体平衡方程（PBE），其计算成本十分高昂。但通过以矩的形式表征演化的粒子密度函数，可有效降低计算开销。其中，基于正交的矩方法（QBMMs）通过正交规则对上述矩进行反演，以近似所需的统计量。已有研究表明，QBMMs能够借助相对紧凑的矩集对喷雾与炭烟实现精准建模。然而，针对气泡动力学这类显著非高斯过程扩展矩集时，该方法会出现数值不稳定问题。针对该问题，本文提出通过训练循环神经网络（RNN）调整QBMM的正交配置，以更高精度求解未闭合矩。所提方法在针对随时间波动压力场响应的气泡振荡简化模型上开展了测试。与传统QBMMs相比，该方法将模型形式误差降低了一个数量级；且由于未扩展基准矩集，其兼具数值稳定性与计算高效性。此外，本文还对额外正交配置点进行了评估，通过额外循环神经网络实现其最优布置与加权。由于矩集仍未改变，这类额外配置点能够以极低的成本进一步降低建模误差。本文属于"动态系统中的数据驱动预测（Data-driven prediction in dynamical systems）"专题议题。