Physics-informed neural networks with multistage augmented Lagrangian terms: Applications to coaxial bubble coalescence in two-phase flow

The study of bubble dynamics in gas-liquid two-phase flow plays an important role in such fields as chemical engineering, hydraulic engineering, and aerospace. In this research, we consider the multibubble coalescence behaviors in still water, based on a machine learning approach combining data with physical constraints, namely, physics-informed neural networks (PINNs). Given the significant changes of different physical quantities including the velocity, pressure, and volume fraction during coaxial bubble coalescence, we develop a modified PINNs framework with multistage augmented Lagrangian terms (MSAL-PINNs) to capture the process of bubble rising, merging, and breaking. We first consider the corresponding penalty terms in different stages to improve the prediction accuracy of the model. We simulate the scenario of double-bubble coalescence. We further extend the cases to the breakup, different radii, and multibubble interactions. Finally, we explore the performance of the model to extrapolate in the parameter domain (σ) as well as the time domain. The results show that the MSAL-PINNs perform well in both aspects. The proposed framework in solving bubble dynamics is validated at both qualitative and quantitative levels, through comparing with computational fluid dynamics and related PINN methods. Our results provide new insights for the intelligent exploration of complex bubble dynamics in gas-liquid two-phase flow.