Discrete-time Physics-Informed Neural Networks for Two-Phase Flow Interface Capturing

In two-phase flow, accurately capturing sharp interfaces is essential. Physics-Informed Neural Networks (PINNs) provide a new way to capture interfaces. This paper introduces a new framework that uses Discrete-time Physics-Informed Neural Networks (DtPINNs) to solve the volume of fluid (VOF) advection equation, offering a new approach to interface capturing in two-phase flows. With this framework, we propose an adaptive collocation-point refinement algorithm, which improves the precision of capturing sharp interfaces, reducing errors and diffusion. Experiments, including cases with translation and deformation, show that the DtPINNs method maintains sharp interfaces, outperforming traditional numerical methods. In a 2D single-vortex deformation case, DtPINNs achieved a relative \({L}_2\) error of 0.0760, much lower than the 5th-order WENO-JS scheme (0.5534) and the 1st-order Upwind scheme (0.8879). This shows that DtPINNs are better at capturing complex interface shapes while keeping accuracy and minimizing numerical diffusion.