Modeling complex fluid flow using machine learning is increasingly recognized as a valuable approach for revealing multiphase fluid phenomena. Bubble dynamics represent a classical two-phase flow problem that plays a crucial role in various engineering domains. In this paper, physics-informed neural networks (PINNs) are applied to facilitate incompressible two-phase bubble motion modeling by integrating governing equations and interface evolution equations. The loss function of PINNs consists of multiple loss terms, including initial and boundary conditions constraints, partial differential equations residuals, and volume fraction constraints. The performance of PINNs is influenced by the competing effects of these loss terms. Therefore, we introduce a heuristic adaptive weights approach to automatically adjust loss weights for each training point, avoiding manual tuning and improving the accuracy of PINNs. We investigate typical bubble motion cases, specifically focusing on bubble rising and breakup, to showcase the capabilities of the proposed method. We explore the impact of weights and present the results in comparison to the baselines. Through the bubble breakup case, we illustrate that our model shows superior performance even with more complex scenarios. Then we further discuss the generalization and robustness of our model, showing their indispensability over traditional solvers in gas–liquid two-phase systems. Specifically, we accelerate computation speed in transfer learning without the need to modify the original model. We also show that our method effectively solves ill-posed problems, such as those without initial data or with incomplete or noisy boundary conditions.