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Fluid dynamics computations for tube-like geometries are important for biomedical evaluation of vascular and airway fluid dynamics. Physics-Informed Neural Networks (PINNs) have recently emerged as a good alternative to traditional computational fluid dynamics (CFD) methods. The vanilla PINN, however, requires much longer training time than the traditional CFD methods for each specific flow scenario and thus does not justify its mainstream use. Here, we explore the use of the multi-case PINN approach for calculating biomedical tube flows, where varied geometry cases are parameterized and pre-trained on the PINN, such that results for unseen geometries can be obtained in real time. Our objective is to identify network architecture, tube-specific, and regularization strategies that can optimize this, via experiments on a series of idealized 2D stenotic tube flows., Comment: 24 pages, 8 figures, 5 tables