We propose physics-informed neural networks (PINNs) as a numerical solver for relativistic spin hydrodynamics and demonstrate that the total angular momentum, i.e., the sum of orbital and spin angular momentum, is accurately conserved throughout the fluid evolution by imposing the conservation law directly in the loss function as a training target. This enables controlled numerical studies of the mutual conversion between spin and orbital angular momentum, a central feature of relativistic spin hydrodynamics driven by the rotational viscous effect. We present two physical scenarios for a rotating fluid confined in a cylindrical container: one case in which initial orbital angular momentum is converted into spin angular momentum in analogy with the Barnett effect, and the opposite case in which initial spin angular momentum is converted into orbital angular momentum in analogy with the Einstein-de Haas effect. Assuming translational invariance along the axial direction and a rotational symmetry, we reduce the dynamics to an effective (1+1)-dimensional problem in time and radial coordinates and investigate these conversion processes governed by the rotational viscous effect by analyzing the spacetime profiles of thermal vorticity and spin potential. Our PINNs-based framework provides the first numerical evidence for spin-orbit angular momentum conversion with fully nonlinear computational relativistic spin hydrodynamics.