Industrial manufacturing processes are often described by systems of differential equations with respective material laws. Since these material laws depend on material-specific parameters, which are frequently unknown or can only be obtained through a range of expensive experiments, there is a need for numerical parameter estimation strategies. In this context, we consider a fiber spinning process modeled by a boundary value problem (BVP) of ordinary differential equations where some material parameters need to be estimated on the basis of few experimental data. We have developed a numerical method involving collocation and continuation methods to solve BVPs arising in a generalized Newtonian setting. Furthermore we are able to calculate gradients of the BVP solutions with respect to the material parameters and apply nonlinear optimization techniques to optimize the material parameters of interest with respect to given measurement data. In this talk we aim to replace the computationally demanding BVPs through surrogate models, leading to fast online optimization. Furthermore we investigate the training process of these surrogates and transfer learning approaches for different fiber spinning configurations.