In this work, we present a physics-informed neural network framework for multiphase flow problems, which accommodates various interface capturing schemes, including phase-field methods and level-set methods. We first demonstrate that for a rotating motion of a circular bubble, the PINN framework with the transport equation, along with the conservative Allen-Cahn terms, captures the interface of the rotating bubble and preserves the shape (mass) of the bubble better than the transport equation alone. Consistent with observations from classical numerical methods, several interface-capturing approaches are employed to avoid sharp discontinuities at the interface; their benefits extend to the PINN setting as well, where they facilitate faster decay of loss terms during the optimization process. Beyond a standard multilayer perceptron (MLP) architecture, we utilized PirateNets, which employ adaptive residual connections that enable the network to be initialized as a shallow model and progressively deepen during training. This adaptive deepening leads to improved predictive accuracy for multiphase flow problems. We further investigate the effectiveness of different approaches proposed in our work in the context of more challenging benchmarks, including the Rider–Kothe vortex and Rayleigh–Taylor instability.