Variations in human body positioning induce rapid changes in venous and arterial flow, as well as swift alterations in blood volume. Veins demonstrate a highly nonlinear pressure-area relation, potentially leading to collapse in sitting or standing humans, unlike arteries. The characterization of these phenomena is crucial in understanding blood flow dynamics. These dynamics can be modeled using one-dimensional approaches, leading to a hyperbolic system of equations with source terms. Their numerical discretization, considering the spatial variation of mechanical and geometrical properties, requires advanced numerical solvers for stability and accurate vessel wall description. The formulation includes geometric-type source terms representing blood discharge, pressure changes, and vessel area. Incorrect handling of these terms can cause spurious oscillations in the numerical solution, leading to prediction failure. This work explores the use of Physics-Informed Neural Networks (PINNs) in conjunction with the mentioned numerical solvers, for an accurate and robust description of nonlinear wave propagation in vessels with geometric-type source terms. Classical PINNs, which embed differential equations governing blood flow and use automatic differentiation for partial derivatives, have shown promise in predicting arterial blood pressure from non-invasive 4D flow MRI data. However, they struggle to accurately reproduce pressure and flow waves, particularly secondary waves. Advanced numerical methods, which can capture these effects more reliably, are combined with PINNs in this study. This hybrid approach uses numerical differentiation instead of traditional automatic differentiation for partial derivative approximation. By including support points around collocation points, it greatly enhances the robustness and generalization capacity of the PINNs. Additionally, incorporating the numerical solver allows the model to capture secondary effects.