In this study, we present a novel Physics-Informed Neural Network (PINN) framework that reconstructs 3D flows using planar velocity projections from arbitrarily oriented planes. The method is designed for the reconstruction of low-Reynolds-number flows typical of cerebrospinal fluid (CSF) in the subarachnoid space (SAS). The method utilizes a projection loss function combined with gradient smoothing regularization during network training. We show that plane orientation with perturbations of 0.05 (relative to the main flow axis) or greater is sufficient axial data for accurate reconstruction. Additionally, gradient exponential moving average smoothing with amplification improves convergence and stability, particularly for near-parallel planes of acquisition. The method was compared against computational fluid dynamics (CFD) data and applied to flow in a realistic canine SAS geometry derived from intravascular optical coherence tomography (OCT), demonstrating the framework's potential for in-vivo CSF flow imaging.