Physics-Informed Neural Networks (PINNs) offer a promising approach to solvingdifferential equations and, more generally, to applying deep learning to problemsin the physical sciences. We adopt a recently developed transfer learning approachfor PINNs and introduce a multi-head model to efficiently obtain accurate solutionsto nonlinear systems of differential equations. In particular, we apply the methodto simulate stochastic branched flows, a universal phenomenon in random wavedynamics. We compare the results achieved by feed forward and GAN-basedPINNs on two physically relevant transfer learning tasks and show that our methodsprovide significant computational speedups in comparison to standard PINNstrained from scratch.