HyFluid is a neural method for inferring fluid density and velocity fields from sparse, multi-view videos. Unlike existing neuro-fluid dynamics reconstruction methods, HyFluid accurately estimates density and reveals the underlying velocity, overcoming the inherent visual blurriness of fluid velocity. The method achieves physically plausible velocity field inference by introducing a set of physics-based losses, while handling the turbulent nature of fluid velocity. It employs a hybrid neural velocity representation, including a base neural velocity field that captures most of the irrotational energy and vortex particle velocities that simulate the remaining turbulent velocity. The method can be used for various learning and reconstruction applications surrounding 3D incompressible flows, including fluid resimulation and editing, future prediction, and neural dynamic scene synthesis.