Medical imaging and numerical simulation of partial differential equations (PDEs) representing biophysical processes, have been combined in the past few decades to provide noninvasive diagnostic and treatment prediction tools for various diseases. Most approaches involve solving computationally expensive PDEs, which can hinder their effective deployment in clinical settings. To overcome this limitation, deep learning has emerged as a promising method to accelerate numerical solvers. One challenge persists however in the generalization abilities of these models, given the wide variety of patient morphologies. This study addresses this challenge by introducing a physics-informed graph neural network designed to solve Darcy equations for the simulation of myocardial perfusion. Leveraging a finite volume discretization of the equations as a "physics-informed" loss, our model was successfully trained and tested on a 3D synthetic dataset, namely meshes representing simplified myocardium shapes. Subsequent evaluation on genuine myocardium meshes, extracted from patients Computed Tomography images, demonstrated promising results, and generalized capabilities. Such a fast solver, within a differentiable learning framework, will enable to tackle inverse problems based on H 2 O-PET perfusion imaging data.